Ore modules¶

Let $R$ be a commutative ring, $θ : K \to K$ by a ring endomorphism and $\partial : K \to K$ be a $θ$ -derivation, that is an additive map satisfying the following axiom

\partial (x y) = θ (x) \partial (y) + \partial (x) y

A Ore module over $(R, θ, \partial)$ is a $R$ -module $M$ equipped with a additive $f : M \to M$ such that

f (a x) = θ (a) f (x) + \partial (a) x

Such a map $f$ is called a pseudomorphism.

Equivalently, a Ore module is a module over the (noncommutative) Ore polynomial ring $S = R [X; θ, \partial]$ .

Defining Ore modules

SageMath provides support for creating and manipulating Ore modules that are finite free over the base ring $R$ .

To start with, the method sage.rings.polynomial.ore_polynomial_ring.OrePolynomialRing.quotient_module() creates the quotient $S / S P$ , endowed with its structure of $S$ -module, that is its structure of Ore module:

Sage

sage: K.<z> = GF(5^3)
sage: S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
sage: M = S.quotient_module(X^2 + z)
sage: M
Ore module of rank 2 over Finite Field in z of size 5^3 twisted by z |--> z^5

Python

>>> from sage.all import *
>>> K = GF(Integer(5)**Integer(3), names=('z',)); (z,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.frobenius_endomorphism(), names=('X',)); (X,) = S._first_ngens(1)
>>> M = S.quotient_module(X**Integer(2) + z)
>>> M
Ore module of rank 2 over Finite Field in z of size 5^3 twisted by z |--> z^5

Sage Live

K.<z> = GF(5^3)
S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
M = S.quotient_module(X^2 + z)
M

Classical methods are available and we can work with elements in $M$ as we do usually for vectors in finite free modules:

Sage

sage: M.basis()
[(1, 0), (0, 1)]

sage: v = M((z, z^2)); v
(z, z^2)
sage: z*v
(z^2, 2*z + 2)

Python

>>> from sage.all import *
>>> M.basis()
[(1, 0), (0, 1)]

>>> v = M((z, z**Integer(2))); v
(z, z^2)
>>> z*v
(z^2, 2*z + 2)

Sage Live

M.basis()
v = M((z, z^2)); v
z*v

The Ore action (or equivalently the structure of $S$ -module) is also easily accessible:

Sage

sage: X*v
(3*z^2 + 2*z, 2*z^2 + 4*z + 4)

Python

>>> from sage.all import *
>>> X*v
(3*z^2 + 2*z, 2*z^2 + 4*z + 4)

Sage Live

X*v

The method sage.modules.ore_module.OreModule.pseudohom() returns the map $f$ defining the action of $X$ :

Sage

sage: M.pseudohom()
Free module pseudomorphism (twisted by z |--> z^5) defined by the matrix
[  0   1]
[4*z   0]
Domain: Ore module of rank 2 over Finite Field in z of size 5^3 twisted by z |--> z^5
Codomain: Ore module of rank 2 over Finite Field in z of size 5^3 twisted by z |--> z^5

Python

>>> from sage.all import *
>>> M.pseudohom()
Free module pseudomorphism (twisted by z |--> z^5) defined by the matrix
[  0   1]
[4*z   0]
Domain: Ore module of rank 2 over Finite Field in z of size 5^3 twisted by z |--> z^5
Codomain: Ore module of rank 2 over Finite Field in z of size 5^3 twisted by z |--> z^5

Sage Live

M.pseudohom()

A useful feature is the possibility to give chosen names to the vectors of the canonical basis. This is easily done as follows:

Sage

sage: N.<u,v,w> = S.quotient_module(X^3 + z*X + 1)
sage: N
Ore module <u, v, w> over Finite Field in z of size 5^3 twisted by z |--> z^5
sage: N.basis()
[u, v, w]

Python

>>> from sage.all import *
>>> N = S.quotient_module(X**Integer(3) + z*X + Integer(1), names=('u', 'v', 'w',)); (u, v, w,) = N._first_ngens(3)
>>> N
Ore module <u, v, w> over Finite Field in z of size 5^3 twisted by z |--> z^5
>>> N.basis()
[u, v, w]

Sage Live

N.<u,v,w> = S.quotient_module(X^3 + z*X + 1)
N
N.basis()

Alternatively, one can pass in the argument names; this could be useful in particular when we want to name the vectors basis $e_{0}, e_{1}, \dots$ :

Sage

sage: A = S.quotient_module(X^11 + z, names='e')
sage: A
Ore module <e0, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10> over Finite Field in z of size 5^3 twisted by z |--> z^5
sage: A.basis()
[e0, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10]

Python

>>> from sage.all import *
>>> A = S.quotient_module(X**Integer(11) + z, names='e')
>>> A
Ore module <e0, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10> over Finite Field in z of size 5^3 twisted by z |--> z^5
>>> A.basis()
[e0, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10]

Sage Live

A = S.quotient_module(X^11 + z, names='e')
A
A.basis()

Do not forget to use the method inject_variables() to get the $e_{i}$ in your namespace:

Sage

sage: e0
Traceback (most recent call last):
...
NameError: name 'e0' is not defined
sage: A.inject_variables()
Defining e0, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10
sage: e0
e0

Python

>>> from sage.all import *
>>> e0
Traceback (most recent call last):
...
NameError: name 'e0' is not defined
>>> A.inject_variables()
Defining e0, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10
>>> e0
e0

Sage Live

e0
A.inject_variables()
e0

Submodules and quotients

SageMath provides facilities for creating submodules and quotient modules of Ore modules. First of all, we define the Ore module $S / S P^{2}$ (for some Ore polynomials $P$ ), which is obviously not simple:

Sage

sage: P = X^2 + z*X + 1
sage: U = S.quotient_module(P^2, names='u')
sage: U.inject_variables()
Defining u0, u1, u2, u3

Python

>>> from sage.all import *
>>> P = X**Integer(2) + z*X + Integer(1)
>>> U = S.quotient_module(P**Integer(2), names='u')
>>> U.inject_variables()
Defining u0, u1, u2, u3

Sage Live

P = X^2 + z*X + 1
U = S.quotient_module(P^2, names='u')
U.inject_variables()

We now build the submodule $S P / S P^{2}$ using the method sage.modules.ore_module.OreModule.span():

Sage

sage: V = U.span(P*u0)
sage: V
Ore module of rank 2 over Finite Field in z of size 5^3 twisted by z |--> z^5
sage: V.basis()
[u0 + (z^2+2*z+2)*u2 + 4*z*u3,
 u1 + (2*z^2+4*z+4)*u2 + u3]

Python

>>> from sage.all import *
>>> V = U.span(P*u0)
>>> V
Ore module of rank 2 over Finite Field in z of size 5^3 twisted by z |--> z^5
>>> V.basis()
[u0 + (z^2+2*z+2)*u2 + 4*z*u3,
 u1 + (2*z^2+4*z+4)*u2 + u3]

Sage Live

V = U.span(P*u0)
V
V.basis()

We underline that the span is really the $S$ -span and not the $R$ -span (as otherwise, it will not be a Ore module).

As before, one can use the attributes names to give explicit names to the basis vectors:

Sage

sage: V = U.span(P*u0, names='v')
sage: V
Ore module <v0, v1> over Finite Field in z of size 5^3 twisted by z |--> z^5
sage: V.inject_variables()
Defining v0, v1
sage: v0
v0
sage: U(v0)
u0 + (z^2+2*z+2)*u2 + 4*z*u3

Python

>>> from sage.all import *
>>> V = U.span(P*u0, names='v')
>>> V
Ore module <v0, v1> over Finite Field in z of size 5^3 twisted by z |--> z^5
>>> V.inject_variables()
Defining v0, v1
>>> v0
v0
>>> U(v0)
u0 + (z^2+2*z+2)*u2 + 4*z*u3

Sage Live

V = U.span(P*u0, names='v')
V
V.inject_variables()
v0
U(v0)

A coercion map from $V$ to $U$ is automatically created. Hence, we can safely combine vectors in $V$ and vectors in $U$ in a single expression:

Sage

sage: v0 - u0
(z^2+2*z+2)*u2 + 4*z*u3

Python

>>> from sage.all import *
>>> v0 - u0
(z^2+2*z+2)*u2 + 4*z*u3

Sage Live

v0 - u0

We can create the quotient $U / V$ using a similar syntax:

Sage

sage: W = U.quo(P*u0)
sage: W
Ore module of rank 2 over Finite Field in z of size 5^3 twisted by z |--> z^5
sage: W.basis()
[u2, u3]

Python

>>> from sage.all import *
>>> W = U.quo(P*u0)
>>> W
Ore module of rank 2 over Finite Field in z of size 5^3 twisted by z |--> z^5
>>> W.basis()
[u2, u3]

Sage Live

W = U.quo(P*u0)
W
W.basis()

We see that SageMath reuses by default the names of the representatives to denote the vectors in the quotient $U / V$ . This behaviour can be overridden by providing explicit names using the attributes names.

Shortcuts for creating quotients are also available:

Sage

sage: U / (P*u0)
Ore module of rank 2 over Finite Field in z of size 5^3 twisted by z |--> z^5
sage: U/V
Ore module of rank 2 over Finite Field in z of size 5^3 twisted by z |--> z^5

Python

>>> from sage.all import *
>>> U / (P*u0)
Ore module of rank 2 over Finite Field in z of size 5^3 twisted by z |--> z^5
>>> U/V
Ore module of rank 2 over Finite Field in z of size 5^3 twisted by z |--> z^5

Sage Live

U / (P*u0)
U/V

Morphisms of Ore modules

For a tutorial on morphisms of Ore modules, we refer to sage.modules.ore_module_morphism.

AUTHOR:

Xavier Caruso (2024-10)

class sage.modules.ore_module.OreAction[source]¶

Bases: Action

Action by left multiplication of Ore polynomial rings over Ore modules.

class sage.modules.ore_module.OreModule(mat, ore, names, category)[source]¶

Bases: UniqueRepresentation, FreeModule_ambient

Generic class for Ore modules.

Element[source]¶: alias of OreModuleElement

basis()[source]¶

Return the canonical basis of this Ore module.

EXAMPLES:

Sage

sage: K.<z> = GF(5^3)
sage: S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
sage: M = S.quotient_module(X^3 - z)
sage: M.basis()
[(1, 0, 0), (0, 1, 0), (0, 0, 1)]

Python

>>> from sage.all import *
>>> K = GF(Integer(5)**Integer(3), names=('z',)); (z,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.frobenius_endomorphism(), names=('X',)); (X,) = S._first_ngens(1)
>>> M = S.quotient_module(X**Integer(3) - z)
>>> M.basis()
[(1, 0, 0), (0, 1, 0), (0, 0, 1)]

Sage Live

K.<z> = GF(5^3)
S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
M = S.quotient_module(X^3 - z)
M.basis()

gen(i)[source]¶

Return the $i$ -th vector of the canonical basis of this Ore module.

EXAMPLES:

Sage

sage: K.<z> = GF(5^3)
sage: S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
sage: M = S.quotient_module(X^3 - z)
sage: M.gen(0)
(1, 0, 0)
sage: M.gen(1)
(0, 1, 0)
sage: M.gen(2)
(0, 0, 1)
sage: M.gen(3)
Traceback (most recent call last):
...
IndexError: generator is not defined

Python

>>> from sage.all import *
>>> K = GF(Integer(5)**Integer(3), names=('z',)); (z,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.frobenius_endomorphism(), names=('X',)); (X,) = S._first_ngens(1)
>>> M = S.quotient_module(X**Integer(3) - z)
>>> M.gen(Integer(0))
(1, 0, 0)
>>> M.gen(Integer(1))
(0, 1, 0)
>>> M.gen(Integer(2))
(0, 0, 1)
>>> M.gen(Integer(3))
Traceback (most recent call last):
...
IndexError: generator is not defined

Sage Live

K.<z> = GF(5^3)
S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
M = S.quotient_module(X^3 - z)
M.gen(0)
M.gen(1)
M.gen(2)
M.gen(3)

gens()[source]¶

Return the canonical basis of this Ore module.

EXAMPLES:

Sage

sage: K.<z> = GF(5^3)
sage: S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
sage: M = S.quotient_module(X^3 - z)
sage: M.gens()
[(1, 0, 0), (0, 1, 0), (0, 0, 1)]

Python

>>> from sage.all import *
>>> K = GF(Integer(5)**Integer(3), names=('z',)); (z,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.frobenius_endomorphism(), names=('X',)); (X,) = S._first_ngens(1)
>>> M = S.quotient_module(X**Integer(3) - z)
>>> M.gens()
[(1, 0, 0), (0, 1, 0), (0, 0, 1)]

Sage Live

K.<z> = GF(5^3)
S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
M = S.quotient_module(X^3 - z)
M.gens()

hom(im_gens, codomain=None)[source]¶

Return the morphism from this Ore module to codomain defined by im_gens.

INPUT:

im_gens – a datum defining the morphism to build; it could either a list, a tuple, a dictionary or a morphism of Ore modules
codomain (default: None) – a Ore module, the codomain of the morphism; if None, it is inferred from im_gens

EXAMPLES:

Sage

sage: K.<t> = Frac(GF(5)['t'])
sage: S.<X> = OrePolynomialRing(K, K.derivation())
sage: P = X^3 + 2*t*X^2 + (t^2 + 2)*X + t
sage: Q = t*X^2 - X + 1

sage: U = S.quotient_module(P, names='u')
sage: U.inject_variables()
Defining u0, u1, u2
sage: V = S.quotient_module(P*Q, names='v')
sage: V.inject_variables()
Defining v0, v1, v2, v3, v4

Python

>>> from sage.all import *
>>> K = Frac(GF(Integer(5))['t'], names=('t',)); (t,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.derivation(), names=('X',)); (X,) = S._first_ngens(1)
>>> P = X**Integer(3) + Integer(2)*t*X**Integer(2) + (t**Integer(2) + Integer(2))*X + t
>>> Q = t*X**Integer(2) - X + Integer(1)

>>> U = S.quotient_module(P, names='u')
>>> U.inject_variables()
Defining u0, u1, u2
>>> V = S.quotient_module(P*Q, names='v')
>>> V.inject_variables()
Defining v0, v1, v2, v3, v4

Sage Live

K.<t> = Frac(GF(5)['t'])
S.<X> = OrePolynomialRing(K, K.derivation())
P = X^3 + 2*t*X^2 + (t^2 + 2)*X + t
Q = t*X^2 - X + 1
U = S.quotient_module(P, names='u')
U.inject_variables()
V = S.quotient_module(P*Q, names='v')
V.inject_variables()

The first method for creating a morphism from $U$ to $V$ is to explicitly write down its matrix in the canonical bases:

Sage

sage: mat = matrix(3, 5, [1, 4, t, 0, 0,
....:                     0, 1, 0, t, 0,
....:                     0, 0, 1, 1, t])
sage: f = U.hom(mat, codomain=V)
sage: f
Ore module morphism:
  From: Ore module <u0, u1, u2> over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt
  To:   Ore module <v0, v1, v2, v3, v4> over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt

Python

>>> from sage.all import *
>>> mat = matrix(Integer(3), Integer(5), [Integer(1), Integer(4), t, Integer(0), Integer(0),
...                     Integer(0), Integer(1), Integer(0), t, Integer(0),
...                     Integer(0), Integer(0), Integer(1), Integer(1), t])
>>> f = U.hom(mat, codomain=V)
>>> f
Ore module morphism:
  From: Ore module <u0, u1, u2> over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt
  To:   Ore module <v0, v1, v2, v3, v4> over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt

Sage Live

mat = matrix(3, 5, [1, 4, t, 0, 0,
                    0, 1, 0, t, 0,
                    0, 0, 1, 1, t])
f = U.hom(mat, codomain=V)
f

This method is however not really convenient because it requires to compute beforehand all the entries of the defining matrix. Instead, we can pass the list of images of the generators:

Sage

sage: g = U.hom([Q*v0, X*Q*v0, X^2*Q*v0])
sage: g
Ore module morphism:
  From: Ore module <u0, u1, u2> over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt
  To:   Ore module <v0, v1, v2, v3, v4> over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt
sage: g.matrix()
[1 4 t 0 0]
[0 1 0 t 0]
[0 0 1 1 t]

Python

>>> from sage.all import *
>>> g = U.hom([Q*v0, X*Q*v0, X**Integer(2)*Q*v0])
>>> g
Ore module morphism:
  From: Ore module <u0, u1, u2> over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt
  To:   Ore module <v0, v1, v2, v3, v4> over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt
>>> g.matrix()
[1 4 t 0 0]
[0 1 0 t 0]
[0 0 1 1 t]

Sage Live

g = U.hom([Q*v0, X*Q*v0, X^2*Q*v0])
g
g.matrix()

One can even give the values of the morphism on a smaller set as soon as the latter generates the domain as Ore module. The syntax uses dictionaries as follows:

Sage

sage: h = U.hom({u0: Q*v0})
sage: h
Ore module morphism:
  From: Ore module <u0, u1, u2> over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt
  To:   Ore module <v0, v1, v2, v3, v4> over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt
sage: g == h
True

Python

>>> from sage.all import *
>>> h = U.hom({u0: Q*v0})
>>> h
Ore module morphism:
  From: Ore module <u0, u1, u2> over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt
  To:   Ore module <v0, v1, v2, v3, v4> over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt
>>> g == h
True

Sage Live

h = U.hom({u0: Q*v0})
h
g == h

Finally im_gens can also be itself a Ore morphism, in which case SageMath tries to cast it into a morphism with the requested domains and codomains. As an example below, we restrict $g$ to a subspace:

Sage

sage: C.<c0,c1> = U.span((X + t)*u0)
sage: gC = C.hom(g)
sage: gC
Ore module morphism:
  From: Ore module <c0, c1> over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt
  To:   Ore module <v0, v1, v2, v3, v4> over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt

sage: g(c0) == gC(c0)
True
sage: g(c1) == gC(c1)
True

Python

>>> from sage.all import *
>>> C = U.span((X + t)*u0, names=('c0', 'c1',)); (c0, c1,) = C._first_ngens(2)
>>> gC = C.hom(g)
>>> gC
Ore module morphism:
  From: Ore module <c0, c1> over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt
  To:   Ore module <v0, v1, v2, v3, v4> over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt

>>> g(c0) == gC(c0)
True
>>> g(c1) == gC(c1)
True

Sage Live

C.<c0,c1> = U.span((X + t)*u0)
gC = C.hom(g)
gC
g(c0) == gC(c0)
g(c1) == gC(c1)

identity_morphism()[source]¶

Return the identity morphism of this Ore module.

EXAMPLES:

Sage

sage: K.<a> = GF(7^5)
sage: S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
sage: M.<u,v> = S.quotient_module(X^2 + a*X + a^2)
sage: id = M.identity_morphism()
sage: id
Ore module endomorphism of Ore module <u, v> over Finite Field in a of size 7^5 twisted by a |--> a^7

sage: id(u)
u
sage: id(v)
v

Python

>>> from sage.all import *
>>> K = GF(Integer(7)**Integer(5), names=('a',)); (a,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.frobenius_endomorphism(), names=('X',)); (X,) = S._first_ngens(1)
>>> M = S.quotient_module(X**Integer(2) + a*X + a**Integer(2), names=('u', 'v',)); (u, v,) = M._first_ngens(2)
>>> id = M.identity_morphism()
>>> id
Ore module endomorphism of Ore module <u, v> over Finite Field in a of size 7^5 twisted by a |--> a^7

>>> id(u)
u
>>> id(v)
v

Sage Live

K.<a> = GF(7^5)
S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
M.<u,v> = S.quotient_module(X^2 + a*X + a^2)
id = M.identity_morphism()
id
id(u)
id(v)

is_zero()[source]¶

Return True if this Ore module is reduced to zero.

EXAMPLES:

Sage

sage: K.<z> = GF(5^3)
sage: S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
sage: M = S.quotient_module(X^2 + z)
sage: M
Ore module of rank 2 over Finite Field in z of size 5^3 twisted by z |--> z^5
sage: M.is_zero()
False

sage: Q = M.quo(M)
sage: Q.is_zero()
True

Python

>>> from sage.all import *
>>> K = GF(Integer(5)**Integer(3), names=('z',)); (z,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.frobenius_endomorphism(), names=('X',)); (X,) = S._first_ngens(1)
>>> M = S.quotient_module(X**Integer(2) + z)
>>> M
Ore module of rank 2 over Finite Field in z of size 5^3 twisted by z |--> z^5
>>> M.is_zero()
False

>>> Q = M.quo(M)
>>> Q.is_zero()
True

Sage Live

K.<z> = GF(5^3)
S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
M = S.quotient_module(X^2 + z)
M
M.is_zero()
Q = M.quo(M)
Q.is_zero()

matrix()[source]¶

Return the matrix giving the action of the Ore variable on this Ore module.

EXAMPLES:

Sage

sage: K.<z> = GF(5^3)
sage: S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
sage: P = X^3 + z*X^2 - z^2*X + (z+2)
sage: M = S.quotient_module(P)
sage: M.matrix()
[      0       1       0]
[      0       0       1]
[4*z + 3     z^2     4*z]

Python

>>> from sage.all import *
>>> K = GF(Integer(5)**Integer(3), names=('z',)); (z,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.frobenius_endomorphism(), names=('X',)); (X,) = S._first_ngens(1)
>>> P = X**Integer(3) + z*X**Integer(2) - z**Integer(2)*X + (z+Integer(2))
>>> M = S.quotient_module(P)
>>> M.matrix()
[      0       1       0]
[      0       0       1]
[4*z + 3     z^2     4*z]

Sage Live

K.<z> = GF(5^3)
S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
P = X^3 + z*X^2 - z^2*X + (z+2)
M = S.quotient_module(P)
M.matrix()

We recognize the companion matrix attached to the Ore polynomial $P$ . This is of course not a coincidence given that the pseudomorphism corresponds to the left multiplication

See also

pseudohom()

module()[source]¶

Return the underlying free module of this Ore module.

EXAMPLES:

Sage

sage: A.<t> = QQ['t']
sage: S.<X> = OrePolynomialRing(A, A.derivation())
sage: M = S.quotient_module(X^3 - t)
sage: M
Ore module of rank 3 over Univariate Polynomial Ring in t over Rational Field twisted by d/dt

sage: M.module()
Ambient free module of rank 3 over the principal ideal domain Univariate Polynomial Ring in t over Rational Field

Python

>>> from sage.all import *
>>> A = QQ['t']; (t,) = A._first_ngens(1)
>>> S = OrePolynomialRing(A, A.derivation(), names=('X',)); (X,) = S._first_ngens(1)
>>> M = S.quotient_module(X**Integer(3) - t)
>>> M
Ore module of rank 3 over Univariate Polynomial Ring in t over Rational Field twisted by d/dt

>>> M.module()
Ambient free module of rank 3 over the principal ideal domain Univariate Polynomial Ring in t over Rational Field

Sage Live

A.<t> = QQ['t']
S.<X> = OrePolynomialRing(A, A.derivation())
M = S.quotient_module(X^3 - t)
M
M.module()

multiplication_map(P)[source]¶

Return the multiplication by $P$ acting on this Ore module.

INPUT:

P – a scalar in the base ring, or a Ore polynomial

EXAMPLES:

Sage

sage: K.<a> = GF(7^5)
sage: S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
sage: P = X^3 + a*X^2 + X - a^2
sage: M = S.quotient_module(P)

Python

>>> from sage.all import *
>>> K = GF(Integer(7)**Integer(5), names=('a',)); (a,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.frobenius_endomorphism(), names=('X',)); (X,) = S._first_ngens(1)
>>> P = X**Integer(3) + a*X**Integer(2) + X - a**Integer(2)
>>> M = S.quotient_module(P)

Sage Live

K.<a> = GF(7^5)
S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
P = X^3 + a*X^2 + X - a^2
M = S.quotient_module(P)

We define the scalar multiplication by an element in the base ring:

Sage

sage: f = M.multiplication_map(3)
sage: f
Ore module endomorphism of Ore module of rank 3 over Finite Field in a of size 7^5 twisted by a |--> a^7
sage: f.matrix()
[3 0 0]
[0 3 0]
[0 0 3]

Python

>>> from sage.all import *
>>> f = M.multiplication_map(Integer(3))
>>> f
Ore module endomorphism of Ore module of rank 3 over Finite Field in a of size 7^5 twisted by a |--> a^7
>>> f.matrix()
[3 0 0]
[0 3 0]
[0 0 3]

Sage Live

f = M.multiplication_map(3)
f
f.matrix()

Be careful that an element in the base ring defines a Ore morphism if and only if it is fixed by the twisting morphisms and killed by the derivation (otherwise the multiplication by this element does not commute with the Ore action). In SageMath, attempting to create the multiplication by an element which does not fulfill these requirements leads to an error:

Sage

sage: M.multiplication_map(a)
Traceback (most recent call last):
...
ValueError: does not define a morphism of Ore modules

Python

>>> from sage.all import *
>>> M.multiplication_map(a)
Traceback (most recent call last):
...
ValueError: does not define a morphism of Ore modules

Sage Live

M.multiplication_map(a)

As soon as it defines a Ore morphism, one can also build the left multiplication by an Ore polynomial:

Sage

sage: g = M.multiplication_map(X^5)
sage: g
Ore module endomorphism of Ore module of rank 3 over Finite Field in a of size 7^5 twisted by a |--> a^7
sage: g.matrix()
[    3*a^4 + 3*a^3 + 6*a^2 + 5*a       4*a^4 + 5*a^3 + 2*a^2 + 6         6*a^4 + 6*a^3 + a^2 + 4]
[                        a^2 + 3 5*a^4 + 5*a^3 + 6*a^2 + 4*a + 1                 a^3 + 5*a^2 + 4]
[6*a^4 + 6*a^3 + 3*a^2 + 3*a + 1         4*a^4 + 2*a^3 + 3*a + 5 6*a^4 + 6*a^3 + 2*a^2 + 5*a + 2]

Python

>>> from sage.all import *
>>> g = M.multiplication_map(X**Integer(5))
>>> g
Ore module endomorphism of Ore module of rank 3 over Finite Field in a of size 7^5 twisted by a |--> a^7
>>> g.matrix()
[    3*a^4 + 3*a^3 + 6*a^2 + 5*a       4*a^4 + 5*a^3 + 2*a^2 + 6         6*a^4 + 6*a^3 + a^2 + 4]
[                        a^2 + 3 5*a^4 + 5*a^3 + 6*a^2 + 4*a + 1                 a^3 + 5*a^2 + 4]
[6*a^4 + 6*a^3 + 3*a^2 + 3*a + 1         4*a^4 + 2*a^3 + 3*a + 5 6*a^4 + 6*a^3 + 2*a^2 + 5*a + 2]

Sage Live

g = M.multiplication_map(X^5)
g
g.matrix()

We check that the characteristic polynomial of $g$ is the reduced norm of the Ore polynomial $P$ we started with (this is a classical property):

Sage

sage: g.charpoly()
x^3 + 4*x^2 + 2*x + 5
sage: P.reduced_norm(var='x')
x^3 + 4*x^2 + 2*x + 5

Python

>>> from sage.all import *
>>> g.charpoly()
x^3 + 4*x^2 + 2*x + 5
>>> P.reduced_norm(var='x')
x^3 + 4*x^2 + 2*x + 5

Sage Live

g.charpoly()
P.reduced_norm(var='x')

ore_ring(names='x', action=True)[source]¶

Return the underlying Ore polynomial ring.

INPUT:

names (default: x) – a string, the name of the variable
action (default: True) – a boolean; if True, an action of the Ore polynomial ring on the Ore module is set

EXAMPLES:

Sage

sage: K.<a> = GF(5^3)
sage: S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
sage: M.<e1,e2> = S.quotient_module(X^2 - a)
sage: M.ore_ring()
Ore Polynomial Ring in x over Finite Field in a of size 5^3 twisted by a |--> a^5

Python

>>> from sage.all import *
>>> K = GF(Integer(5)**Integer(3), names=('a',)); (a,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.frobenius_endomorphism(), names=('X',)); (X,) = S._first_ngens(1)
>>> M = S.quotient_module(X**Integer(2) - a, names=('e1', 'e2',)); (e1, e2,) = M._first_ngens(2)
>>> M.ore_ring()
Ore Polynomial Ring in x over Finite Field in a of size 5^3 twisted by a |--> a^5

Sage Live

K.<a> = GF(5^3)
S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
M.<e1,e2> = S.quotient_module(X^2 - a)
M.ore_ring()

We can use a different variable name:

Sage

sage: M.ore_ring('Y')
Ore Polynomial Ring in Y over Finite Field in a of size 5^3 twisted by a |--> a^5

Python

>>> from sage.all import *
>>> M.ore_ring('Y')
Ore Polynomial Ring in Y over Finite Field in a of size 5^3 twisted by a |--> a^5

Sage Live

M.ore_ring('Y')

Alternatively, one can use the following shortcut:

Sage

sage: T.<Z> = M.ore_ring()
sage: T
Ore Polynomial Ring in Z over Finite Field in a of size 5^3 twisted by a |--> a^5

Python

>>> from sage.all import *
>>> T = M.ore_ring(names=('Z',)); (Z,) = T._first_ngens(1)
>>> T
Ore Polynomial Ring in Z over Finite Field in a of size 5^3 twisted by a |--> a^5

Sage Live

T.<Z> = M.ore_ring()
T

In all the above cases, an action of the returned Ore polynomial ring on $M$ is registered:

Sage

sage: Z*e1
e2
sage: Z*e2
a*e1

Python

>>> from sage.all import *
>>> Z*e1
e2
>>> Z*e2
a*e1

Sage Live

Z*e1
Z*e2

Specifying action=False prevents this to happen:

Sage

sage: T.<U> = M.ore_ring(action=False)
sage: U*e1
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *:
    'Ore Polynomial Ring in U over Finite Field in a of size 5^3 twisted by a |--> a^5' and
    'Ore module <e1, e2> over Finite Field in a of size 5^3 twisted by a |--> a^5'

Python

>>> from sage.all import *
>>> T = M.ore_ring(action=False, names=('U',)); (U,) = T._first_ngens(1)
>>> U*e1
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *:
    'Ore Polynomial Ring in U over Finite Field in a of size 5^3 twisted by a |--> a^5' and
    'Ore module <e1, e2> over Finite Field in a of size 5^3 twisted by a |--> a^5'

Sage Live

T.<U> = M.ore_ring(action=False)
U*e1

pseudohom()[source]¶

Return the pseudomorphism giving the action of the Ore variable on this Ore module.

EXAMPLES:

Sage

sage: K.<z> = GF(5^3)
sage: S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
sage: P = X^3 + z*X^2 - z^2*X + (z+2)
sage: M = S.quotient_module(P)
sage: M.pseudohom()
Free module pseudomorphism (twisted by z |--> z^5) defined by the matrix
[      0       1       0]
[      0       0       1]
[4*z + 3     z^2     4*z]
Domain: Ore module of rank 3 over Finite Field in z of size 5^3 twisted by z |--> z^5
Codomain: Ore module of rank 3 over Finite Field in z of size 5^3 twisted by z |--> z^5

Python

>>> from sage.all import *
>>> K = GF(Integer(5)**Integer(3), names=('z',)); (z,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.frobenius_endomorphism(), names=('X',)); (X,) = S._first_ngens(1)
>>> P = X**Integer(3) + z*X**Integer(2) - z**Integer(2)*X + (z+Integer(2))
>>> M = S.quotient_module(P)
>>> M.pseudohom()
Free module pseudomorphism (twisted by z |--> z^5) defined by the matrix
[      0       1       0]
[      0       0       1]
[4*z + 3     z^2     4*z]
Domain: Ore module of rank 3 over Finite Field in z of size 5^3 twisted by z |--> z^5
Codomain: Ore module of rank 3 over Finite Field in z of size 5^3 twisted by z |--> z^5

Sage Live

K.<z> = GF(5^3)
S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
P = X^3 + z*X^2 - z^2*X + (z+2)
M = S.quotient_module(P)
M.pseudohom()

See also

matrix()

quo(sub, names=None, check=True)[source]¶

Return the quotient of this Ore module by the submodule generated (over the underlying Ore ring) by gens.

INPUT:

gens – a list of vectors or submodules of this Ore module
names (default: None) – the name of the vectors in a basis of the quotient
check (default: True) – a boolean, ignored

EXAMPLES:

Sage

sage: K.<t> = Frac(GF(5)['t'])
sage: S.<X> = OrePolynomialRing(K, K.derivation())
sage: P = X^2 + t*X + 1
sage: M = S.quotient_module(P^3, names='e')
sage: M.inject_variables()
Defining e0, e1, e2, e3, e4, e5

Python

>>> from sage.all import *
>>> K = Frac(GF(Integer(5))['t'], names=('t',)); (t,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.derivation(), names=('X',)); (X,) = S._first_ngens(1)
>>> P = X**Integer(2) + t*X + Integer(1)
>>> M = S.quotient_module(P**Integer(3), names='e')
>>> M.inject_variables()
Defining e0, e1, e2, e3, e4, e5

Sage Live

K.<t> = Frac(GF(5)['t'])
S.<X> = OrePolynomialRing(K, K.derivation())
P = X^2 + t*X + 1
M = S.quotient_module(P^3, names='e')
M.inject_variables()

We create the quotient $M / M P$ :

Sage

sage: modP = M.quotient(P*e0)
sage: modP
Ore module of rank 2 over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt

Python

>>> from sage.all import *
>>> modP = M.quotient(P*e0)
>>> modP
Ore module of rank 2 over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt

Sage Live

modP = M.quotient(P*e0)
modP

As a shortcut, we can write quo instead of quotient or even use the / operator:

Sage

sage: modP = M / (P*e0)
sage: modP
Ore module of rank 2 over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt

Python

>>> from sage.all import *
>>> modP = M / (P*e0)
>>> modP
Ore module of rank 2 over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt

Sage Live

modP = M / (P*e0)
modP

By default, the vectors in the quotient have the same names as their representatives in $M$ :

Sage

sage: modP.basis()
[e4, e5]

Python

>>> from sage.all import *
>>> modP.basis()
[e4, e5]

Sage Live

modP.basis()

One can override this behavior by setting the attributes names:

Sage

sage: modP = M.quo(P*e0, names='u')
sage: modP.inject_variables()
Defining u0, u1
sage: modP.basis()
[u0, u1]

Python

>>> from sage.all import *
>>> modP = M.quo(P*e0, names='u')
>>> modP.inject_variables()
Defining u0, u1
>>> modP.basis()
[u0, u1]

Sage Live

modP = M.quo(P*e0, names='u')
modP.inject_variables()
modP.basis()

Note that a coercion map from the initial Ore module to its quotient is automatically set. As a consequence, combining elements of M and modP in the same formula works:

Sage

sage: t*u0 + e1
(t^3+4*t)*u0 + (t^2+2)*u1

Python

>>> from sage.all import *
>>> t*u0 + e1
(t^3+4*t)*u0 + (t^2+2)*u1

Sage Live

t*u0 + e1

One can combine the construction of quotients and submodules without trouble. For instance, here we build the space $M P / M P^{2}$ :

Sage

sage: modP2 = M / (P^2*e0)
sage: N = modP2.span(P*e0)
sage: N
Ore module of rank 2 over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt
sage: N.basis()
[e2 + (2*t^2+2)*e4 + 2*t*e5,
 e3 + 4*t*e4 + 4*e5]

Python

>>> from sage.all import *
>>> modP2 = M / (P**Integer(2)*e0)
>>> N = modP2.span(P*e0)
>>> N
Ore module of rank 2 over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt
>>> N.basis()
[e2 + (2*t^2+2)*e4 + 2*t*e5,
 e3 + 4*t*e4 + 4*e5]

Sage Live

modP2 = M / (P^2*e0)
N = modP2.span(P*e0)
N
N.basis()

See also

quo(), span()

quotient(sub, names=None, check=True)[source]¶

Return the quotient of this Ore module by the submodule generated (over the underlying Ore ring) by gens.

INPUT:

gens – a list of vectors or submodules of this Ore module
names (default: None) – the name of the vectors in a basis of the quotient
check (default: True) – a boolean, ignored

EXAMPLES:

Sage

sage: K.<t> = Frac(GF(5)['t'])
sage: S.<X> = OrePolynomialRing(K, K.derivation())
sage: P = X^2 + t*X + 1
sage: M = S.quotient_module(P^3, names='e')
sage: M.inject_variables()
Defining e0, e1, e2, e3, e4, e5

Python

>>> from sage.all import *
>>> K = Frac(GF(Integer(5))['t'], names=('t',)); (t,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.derivation(), names=('X',)); (X,) = S._first_ngens(1)
>>> P = X**Integer(2) + t*X + Integer(1)
>>> M = S.quotient_module(P**Integer(3), names='e')
>>> M.inject_variables()
Defining e0, e1, e2, e3, e4, e5

Sage Live

K.<t> = Frac(GF(5)['t'])
S.<X> = OrePolynomialRing(K, K.derivation())
P = X^2 + t*X + 1
M = S.quotient_module(P^3, names='e')
M.inject_variables()

We create the quotient $M / M P$ :

Sage

sage: modP = M.quotient(P*e0)
sage: modP
Ore module of rank 2 over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt

Python

>>> from sage.all import *
>>> modP = M.quotient(P*e0)
>>> modP
Ore module of rank 2 over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt

Sage Live

modP = M.quotient(P*e0)
modP

As a shortcut, we can write quo instead of quotient or even use the / operator:

Sage

sage: modP = M / (P*e0)
sage: modP
Ore module of rank 2 over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt

Python

>>> from sage.all import *
>>> modP = M / (P*e0)
>>> modP
Ore module of rank 2 over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt

Sage Live

modP = M / (P*e0)
modP

By default, the vectors in the quotient have the same names as their representatives in $M$ :

Sage

sage: modP.basis()
[e4, e5]

Python

>>> from sage.all import *
>>> modP.basis()
[e4, e5]

Sage Live

modP.basis()

One can override this behavior by setting the attributes names:

Sage

sage: modP = M.quo(P*e0, names='u')
sage: modP.inject_variables()
Defining u0, u1
sage: modP.basis()
[u0, u1]

Python

>>> from sage.all import *
>>> modP = M.quo(P*e0, names='u')
>>> modP.inject_variables()
Defining u0, u1
>>> modP.basis()
[u0, u1]

Sage Live

modP = M.quo(P*e0, names='u')
modP.inject_variables()
modP.basis()

Note that a coercion map from the initial Ore module to its quotient is automatically set. As a consequence, combining elements of M and modP in the same formula works:

Sage

sage: t*u0 + e1
(t^3+4*t)*u0 + (t^2+2)*u1

Python

>>> from sage.all import *
>>> t*u0 + e1
(t^3+4*t)*u0 + (t^2+2)*u1

Sage Live

t*u0 + e1

One can combine the construction of quotients and submodules without trouble. For instance, here we build the space $M P / M P^{2}$ :

Sage

sage: modP2 = M / (P^2*e0)
sage: N = modP2.span(P*e0)
sage: N
Ore module of rank 2 over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt
sage: N.basis()
[e2 + (2*t^2+2)*e4 + 2*t*e5,
 e3 + 4*t*e4 + 4*e5]

Python

>>> from sage.all import *
>>> modP2 = M / (P**Integer(2)*e0)
>>> N = modP2.span(P*e0)
>>> N
Ore module of rank 2 over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt
>>> N.basis()
[e2 + (2*t^2+2)*e4 + 2*t*e5,
 e3 + 4*t*e4 + 4*e5]

Sage Live

modP2 = M / (P^2*e0)
N = modP2.span(P*e0)
N
N.basis()

See also

quo(), span()

random_element(*args, **kwds)[source]¶

Return a random element in this Ore module.

Extra arguments are passed to the random generator of the base ring.

EXAMPLES:

Sage

sage: A.<t> = QQ['t']
sage: S.<X> = OrePolynomialRing(A, A.derivation())
sage: M = S.quotient_module(X^3 - t, names='e')
sage: M.random_element()   # random
(-1/2*t^2 - 3/4*t + 3/2)*e0 + (-3/2*t^2 - 3*t + 4)*e1 + (-6*t + 2)*e2

sage: M.random_element(degree=5)   # random
(4*t^5 - 1/2*t^4 + 3/2*t^3 + 6*t^2 - t - 1/10)*e0 + (19/3*t^5 - t^3 - t^2 + 1)*e1 + (t^5 + 4*t^4 + 4*t^2 + 1/3*t - 33)*e2

Python

>>> from sage.all import *
>>> A = QQ['t']; (t,) = A._first_ngens(1)
>>> S = OrePolynomialRing(A, A.derivation(), names=('X',)); (X,) = S._first_ngens(1)
>>> M = S.quotient_module(X**Integer(3) - t, names='e')
>>> M.random_element()   # random
(-1/2*t^2 - 3/4*t + 3/2)*e0 + (-3/2*t^2 - 3*t + 4)*e1 + (-6*t + 2)*e2

>>> M.random_element(degree=Integer(5))   # random
(4*t^5 - 1/2*t^4 + 3/2*t^3 + 6*t^2 - t - 1/10)*e0 + (19/3*t^5 - t^3 - t^2 + 1)*e1 + (t^5 + 4*t^4 + 4*t^2 + 1/3*t - 33)*e2

Sage Live

A.<t> = QQ['t']
S.<X> = OrePolynomialRing(A, A.derivation())
M = S.quotient_module(X^3 - t, names='e')
M.random_element()   # random
M.random_element(degree=5)   # random

rename_basis(names, coerce=False)[source]¶

Return the same Ore module with the given naming for the vectors in its distinguished basis.

INPUT:

names – a string or a list of strings, the new names
coerce (default: False) – a boolean; if True, a coercion map from this Ore module to renamed version is set

EXAMPLES:

Sage

sage: K.<z> = GF(5^3)
sage: S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
sage: M = S.quotient_module(X^2 + z)
sage: M
Ore module of rank 2 over Finite Field in z of size 5^3 twisted by z |--> z^5

sage: Me = M.rename_basis('e')
sage: Me
Ore module <e0, e1> over Finite Field in z of size 5^3 twisted by z |--> z^5

Python

>>> from sage.all import *
>>> K = GF(Integer(5)**Integer(3), names=('z',)); (z,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.frobenius_endomorphism(), names=('X',)); (X,) = S._first_ngens(1)
>>> M = S.quotient_module(X**Integer(2) + z)
>>> M
Ore module of rank 2 over Finite Field in z of size 5^3 twisted by z |--> z^5

>>> Me = M.rename_basis('e')
>>> Me
Ore module <e0, e1> over Finite Field in z of size 5^3 twisted by z |--> z^5

Sage Live

K.<z> = GF(5^3)
S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
M = S.quotient_module(X^2 + z)
M
Me = M.rename_basis('e')
Me

Now compare how elements are displayed:

Sage

sage: M.random_element()   # random
(3*z^2 + 4*z + 2, 3*z^2 + z)
sage: Me.random_element()  # random
(2*z+4)*e0 + (z^2+4*z+4)*e1

Python

>>> from sage.all import *
>>> M.random_element()   # random
(3*z^2 + 4*z + 2, 3*z^2 + z)
>>> Me.random_element()  # random
(2*z+4)*e0 + (z^2+4*z+4)*e1

Sage Live

M.random_element()   # random
Me.random_element()  # random

At this point, there is no coercion map between M and Me. Therefore, adding elements in both parents results in an error:

Sage

sage: M.random_element() + Me.random_element()
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for +:
'Ore module of rank 2 over Finite Field in z of size 5^3 twisted by z |--> z^5' and
'Ore module <e0, e1> over Finite Field in z of size 5^3 twisted by z |--> z^5'

Python

>>> from sage.all import *
>>> M.random_element() + Me.random_element()
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for +:
'Ore module of rank 2 over Finite Field in z of size 5^3 twisted by z |--> z^5' and
'Ore module <e0, e1> over Finite Field in z of size 5^3 twisted by z |--> z^5'

Sage Live

M.random_element() + Me.random_element()

In order to set this coercion, one should define Me by passing the extra argument coerce=True:

Sage

sage: Me = M.rename_basis('e', coerce=True)
sage: M.random_element() + Me.random_element()  # random
2*z^2*e0 + (z^2+z+4)*e1

Python

>>> from sage.all import *
>>> Me = M.rename_basis('e', coerce=True)
>>> M.random_element() + Me.random_element()  # random
2*z^2*e0 + (z^2+z+4)*e1

Sage Live

Me = M.rename_basis('e', coerce=True)
M.random_element() + Me.random_element()  # random

Warning

Use coerce=True with extreme caution. Indeed, setting inappropriate coercion maps may result in a circular path in the coercion graph which, in turn, could eventually break the coercion system.

Note that the bracket construction also works:

Sage

sage: M.<v,w> = M.rename_basis()
sage: M
Ore module <v, w> over Finite Field in z of size 5^3 twisted by z |--> z^5

Python

>>> from sage.all import *
>>> M = M.rename_basis(names=('v', 'w',)); (v, w,) = M._first_ngens(2)
>>> M
Ore module <v, w> over Finite Field in z of size 5^3 twisted by z |--> z^5

Sage Live

M.<v,w> = M.rename_basis()
M

In this case, $v$ and $w$ are automatically defined:

Sage

sage: v + w
v + w

Python

>>> from sage.all import *
>>> v + w
v + w

Sage Live

v + w

span(gens, names=None)[source]¶

Return the submodule of this Ore module generated (over the underlying Ore ring) by gens.

INPUT:

gens – a list of vectors or submodules of this Ore module
names (default: None) – the name of the vectors in a basis of this submodule

EXAMPLES:

Sage

sage: K.<t> = Frac(GF(5)['t'])
sage: S.<X> = OrePolynomialRing(K, K.derivation())
sage: P = X^2 + t*X + 1
sage: M = S.quotient_module(P^3, names='e')
sage: M.inject_variables()
Defining e0, e1, e2, e3, e4, e5

Python

>>> from sage.all import *
>>> K = Frac(GF(Integer(5))['t'], names=('t',)); (t,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.derivation(), names=('X',)); (X,) = S._first_ngens(1)
>>> P = X**Integer(2) + t*X + Integer(1)
>>> M = S.quotient_module(P**Integer(3), names='e')
>>> M.inject_variables()
Defining e0, e1, e2, e3, e4, e5

Sage Live

K.<t> = Frac(GF(5)['t'])
S.<X> = OrePolynomialRing(K, K.derivation())
P = X^2 + t*X + 1
M = S.quotient_module(P^3, names='e')
M.inject_variables()

We create the submodule $M P$ :

Sage

sage: MP = M.span([P*e0])
sage: MP
Ore module of rank 4 over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt
sage: MP.basis()
[e0 + (t^4+t^2+3)*e4 + t^3*e5,
 e1 + (4*t^3+2*t)*e4 + (4*t^2+3)*e5,
 e2 + (2*t^2+2)*e4 + 2*t*e5,
 e3 + 4*t*e4 + 4*e5]

Python

>>> from sage.all import *
>>> MP = M.span([P*e0])
>>> MP
Ore module of rank 4 over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt
>>> MP.basis()
[e0 + (t^4+t^2+3)*e4 + t^3*e5,
 e1 + (4*t^3+2*t)*e4 + (4*t^2+3)*e5,
 e2 + (2*t^2+2)*e4 + 2*t*e5,
 e3 + 4*t*e4 + 4*e5]

Sage Live

MP = M.span([P*e0])
MP
MP.basis()

When there is only one generator, encapsulating it in a list is not necessary; one can equally write:

Sage

sage: MP = M.span(P*e0)

Python

>>> from sage.all import *
>>> MP = M.span(P*e0)

Sage Live

MP = M.span(P*e0)

If one wants, one can give names to the basis of the submodule using the attribute names:

Sage

sage: MP2 = M.span(P^2*e0, names='u')
sage: MP2.inject_variables()
Defining u0, u1
sage: MP2.basis()
[u0, u1]

sage: M(u0)
e0 + (t^2+4)*e2 + 3*t^3*e3 + (t^2+1)*e4 + 3*t*e5

Python

>>> from sage.all import *
>>> MP2 = M.span(P**Integer(2)*e0, names='u')
>>> MP2.inject_variables()
Defining u0, u1
>>> MP2.basis()
[u0, u1]

>>> M(u0)
e0 + (t^2+4)*e2 + 3*t^3*e3 + (t^2+1)*e4 + 3*t*e5

Sage Live

MP2 = M.span(P^2*e0, names='u')
MP2.inject_variables()
MP2.basis()
M(u0)

Note that a coercion map from the submodule to the ambient module is automatically set:

Sage

sage: M.has_coerce_map_from(MP2)
True

Python

>>> from sage.all import *
>>> M.has_coerce_map_from(MP2)
True

Sage Live

M.has_coerce_map_from(MP2)

Therefore, combining elements of M and MP2 in the same expression perfectly works:

Sage

sage: t*u0 + e1
t*e0 + e1 + (t^3+4*t)*e2 + 3*t^4*e3 + (t^3+t)*e4 + 3*t^2*e5

Python

>>> from sage.all import *
>>> t*u0 + e1
t*e0 + e1 + (t^3+4*t)*e2 + 3*t^4*e3 + (t^3+t)*e4 + 3*t^2*e5

Sage Live

t*u0 + e1

Here is an example with multiple generators:

Sage

sage: MM = M.span([MP2, P*e1])
sage: MM.basis()
[e0, e1, e2, e3, e4, e5]

Python

>>> from sage.all import *
>>> MM = M.span([MP2, P*e1])
>>> MM.basis()
[e0, e1, e2, e3, e4, e5]

Sage Live

MM = M.span([MP2, P*e1])
MM.basis()

In this case, we obtain the whole space.

Creating submodules of submodules is also allowed:

Sage

sage: N = MP.span(P^2*e0)
sage: N
Ore module of rank 2 over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt
sage: N.basis()
[e0 + (t^2+4)*e2 + 3*t^3*e3 + (t^2+1)*e4 + 3*t*e5,
 e1 + (4*t^2+4)*e3 + 3*t*e4 + 4*e5]

Python

>>> from sage.all import *
>>> N = MP.span(P**Integer(2)*e0)
>>> N
Ore module of rank 2 over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt
>>> N.basis()
[e0 + (t^2+4)*e2 + 3*t^3*e3 + (t^2+1)*e4 + 3*t*e5,
 e1 + (4*t^2+4)*e3 + 3*t*e4 + 4*e5]

Sage Live

N = MP.span(P^2*e0)
N
N.basis()

See also

quotient()

twisting_derivation()[source]¶

Return the twisting derivation corresponding to this Ore module.

EXAMPLES:

Sage

sage: R.<t> = QQ[]
sage: T.<Y> = OrePolynomialRing(R, R.derivation())
sage: M = T.quotient_module(Y + t^2)
sage: M.twisting_derivation()
d/dt

Python

>>> from sage.all import *
>>> R = QQ['t']; (t,) = R._first_ngens(1)
>>> T = OrePolynomialRing(R, R.derivation(), names=('Y',)); (Y,) = T._first_ngens(1)
>>> M = T.quotient_module(Y + t**Integer(2))
>>> M.twisting_derivation()
d/dt

Sage Live

R.<t> = QQ[]
T.<Y> = OrePolynomialRing(R, R.derivation())
M = T.quotient_module(Y + t^2)
M.twisting_derivation()

When the twisting derivation in zero, nothing is returned:

Sage

sage: K.<z> = GF(5^3)
sage: S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
sage: M = S.quotient_module(X + z)
sage: M.twisting_derivation()

Python

>>> from sage.all import *
>>> K = GF(Integer(5)**Integer(3), names=('z',)); (z,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.frobenius_endomorphism(), names=('X',)); (X,) = S._first_ngens(1)
>>> M = S.quotient_module(X + z)
>>> M.twisting_derivation()

Sage Live

K.<z> = GF(5^3)
S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
M = S.quotient_module(X + z)
M.twisting_derivation()

See also

twisting_morphism()

twisting_morphism()[source]¶

Return the twisting morphism corresponding to this Ore module.

EXAMPLES:

Sage

sage: K.<z> = GF(5^3)
sage: S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
sage: M = S.quotient_module(X + z)
sage: M.twisting_morphism()
Frobenius endomorphism z |--> z^5 on Finite Field in z of size 5^3

Python

>>> from sage.all import *
>>> K = GF(Integer(5)**Integer(3), names=('z',)); (z,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.frobenius_endomorphism(), names=('X',)); (X,) = S._first_ngens(1)
>>> M = S.quotient_module(X + z)
>>> M.twisting_morphism()
Frobenius endomorphism z |--> z^5 on Finite Field in z of size 5^3

Sage Live

K.<z> = GF(5^3)
S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
M = S.quotient_module(X + z)
M.twisting_morphism()

When the twisting morphism is trivial (that is, the identity), nothing is returned:

Sage

sage: R.<t> = QQ[]
sage: T.<Y> = OrePolynomialRing(R, R.derivation())
sage: M = T.quotient_module(Y + t^2)
sage: M.twisting_morphism()

Python

>>> from sage.all import *
>>> R = QQ['t']; (t,) = R._first_ngens(1)
>>> T = OrePolynomialRing(R, R.derivation(), names=('Y',)); (Y,) = T._first_ngens(1)
>>> M = T.quotient_module(Y + t**Integer(2))
>>> M.twisting_morphism()

Sage Live

R.<t> = QQ[]
T.<Y> = OrePolynomialRing(R, R.derivation())
M = T.quotient_module(Y + t^2)
M.twisting_morphism()

See also

twisting_derivation()

class sage.modules.ore_module.OreQuotientModule(cover, basis, names)[source]¶

Bases: OreModule

Class for quotients of Ore modules.

cover()[source]¶

If this quotient in $M / N$ , return $M$ .

EXAMPLES:

Sage

sage: K.<t> = Frac(GF(5)['t'])
sage: S.<X> = OrePolynomialRing(K, K.derivation())
sage: M.<v,w> = S.quotient_module((X + t)^2)
sage: N = M.quo((X + t)*v)

sage: N.cover()
Ore module <v, w> over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt
sage: N.cover() is M
True

Python

>>> from sage.all import *
>>> K = Frac(GF(Integer(5))['t'], names=('t',)); (t,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.derivation(), names=('X',)); (X,) = S._first_ngens(1)
>>> M = S.quotient_module((X + t)**Integer(2), names=('v', 'w',)); (v, w,) = M._first_ngens(2)
>>> N = M.quo((X + t)*v)

>>> N.cover()
Ore module <v, w> over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt
>>> N.cover() is M
True

Sage Live

K.<t> = Frac(GF(5)['t'])
S.<X> = OrePolynomialRing(K, K.derivation())
M.<v,w> = S.quotient_module((X + t)^2)
N = M.quo((X + t)*v)
N.cover()
N.cover() is M

See also

relations()

morphism_modulo(f)[source]¶

If this quotient in $M / N$ and $f : X \to M$ is a morphism, return the induced map $X \to M / N$ .

EXAMPLES:

Sage

sage: K.<t> = Frac(GF(5)['t'])
sage: S.<X> = OrePolynomialRing(K, K.derivation())
sage: P = X + t
sage: M.<v,w> = S.quotient_module(P^2)
sage: Q.<wbar> = M.quo(P*v)

sage: f = M.multiplication_map(X^5)
sage: f
Ore module endomorphism of Ore module <v, w> over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt
sage: g = Q.morphism_modulo(f)
sage: g
Ore module morphism:
  From: Ore module <v, w> over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt
  To:   Ore module <wbar> over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt

Python

>>> from sage.all import *
>>> K = Frac(GF(Integer(5))['t'], names=('t',)); (t,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.derivation(), names=('X',)); (X,) = S._first_ngens(1)
>>> P = X + t
>>> M = S.quotient_module(P**Integer(2), names=('v', 'w',)); (v, w,) = M._first_ngens(2)
>>> Q = M.quo(P*v, names=('wbar',)); (wbar,) = Q._first_ngens(1)

>>> f = M.multiplication_map(X**Integer(5))
>>> f
Ore module endomorphism of Ore module <v, w> over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt
>>> g = Q.morphism_modulo(f)
>>> g
Ore module morphism:
  From: Ore module <v, w> over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt
  To:   Ore module <wbar> over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt

Sage Live

K.<t> = Frac(GF(5)['t'])
S.<X> = OrePolynomialRing(K, K.derivation())
P = X + t
M.<v,w> = S.quotient_module(P^2)
Q.<wbar> = M.quo(P*v)
f = M.multiplication_map(X^5)
f
g = Q.morphism_modulo(f)
g

morphism_quotient(f)[source]¶

If this quotient in $M / N$ and $f : M \to X$ is a morphism vanishing on $N$ , return the induced map $M / N \to X$ .

EXAMPLES:

Sage

sage: K.<t> = Frac(GF(5)['t'])
sage: S.<X> = OrePolynomialRing(K, K.derivation())
sage: P = X + t
sage: M.<v,w> = S.quotient_module(P^2)
sage: Q.<wbar> = M.quo(P*v)

sage: f = M.hom({v: P*v})
sage: f
Ore module endomorphism of Ore module <v, w> over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt
sage: g = Q.morphism_quotient(f)
sage: g
Ore module morphism:
  From: Ore module <wbar> over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt
  To:   Ore module <v, w> over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt

Python

>>> from sage.all import *
>>> K = Frac(GF(Integer(5))['t'], names=('t',)); (t,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.derivation(), names=('X',)); (X,) = S._first_ngens(1)
>>> P = X + t
>>> M = S.quotient_module(P**Integer(2), names=('v', 'w',)); (v, w,) = M._first_ngens(2)
>>> Q = M.quo(P*v, names=('wbar',)); (wbar,) = Q._first_ngens(1)

>>> f = M.hom({v: P*v})
>>> f
Ore module endomorphism of Ore module <v, w> over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt
>>> g = Q.morphism_quotient(f)
>>> g
Ore module morphism:
  From: Ore module <wbar> over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt
  To:   Ore module <v, w> over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt

Sage Live

K.<t> = Frac(GF(5)['t'])
S.<X> = OrePolynomialRing(K, K.derivation())
P = X + t
M.<v,w> = S.quotient_module(P^2)
Q.<wbar> = M.quo(P*v)
f = M.hom({v: P*v})
f
g = Q.morphism_quotient(f)
g

When the given morphism does not vanish on $N$ , an error is raised:

Sage

sage: h = M.multiplication_map(X^5)
sage: h
Ore module endomorphism of Ore module <v, w> over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt
sage: Q.morphism_quotient(h)
Traceback (most recent call last):
...
ValueError: the morphism does not factor through this quotient

Python

>>> from sage.all import *
>>> h = M.multiplication_map(X**Integer(5))
>>> h
Ore module endomorphism of Ore module <v, w> over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt
>>> Q.morphism_quotient(h)
Traceback (most recent call last):
...
ValueError: the morphism does not factor through this quotient

Sage Live

h = M.multiplication_map(X^5)
h
Q.morphism_quotient(h)

projection_morphism()[source]¶

Return the projection from the cover module to this quotient.

EXAMPLES:

Sage

sage: K.<t> = Frac(GF(5)['t'])
sage: S.<X> = OrePolynomialRing(K, K.derivation())
sage: M.<v,w> = S.quotient_module((X + t)^2)
sage: Q = M.quo((X + t)*v)
sage: Q.projection_morphism()
Ore module morphism:
  From: Ore module <v, w> over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt
  To:   Ore module of rank 1 over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt

Python

>>> from sage.all import *
>>> K = Frac(GF(Integer(5))['t'], names=('t',)); (t,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.derivation(), names=('X',)); (X,) = S._first_ngens(1)
>>> M = S.quotient_module((X + t)**Integer(2), names=('v', 'w',)); (v, w,) = M._first_ngens(2)
>>> Q = M.quo((X + t)*v)
>>> Q.projection_morphism()
Ore module morphism:
  From: Ore module <v, w> over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt
  To:   Ore module of rank 1 over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt

Sage Live

K.<t> = Frac(GF(5)['t'])
S.<X> = OrePolynomialRing(K, K.derivation())
M.<v,w> = S.quotient_module((X + t)^2)
Q = M.quo((X + t)*v)
Q.projection_morphism()

relations(names=None)[source]¶

If this quotient in $M / N$ , return $N$ .

INPUT:

names – the names of the vectors of the basis of $N$ , or None

EXAMPLES:

Sage

sage: K.<t> = Frac(GF(5)['t'])
sage: S.<X> = OrePolynomialRing(K, K.derivation())
sage: M.<v,w> = S.quotient_module((X + t)^2)
sage: Q = M.quo((X + t)*v)

sage: N = Q.relations()
sage: N
Ore module of rank 1 over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt
sage: (X + t)*v in N
True
sage: Q == M/N
True

Python

>>> from sage.all import *
>>> K = Frac(GF(Integer(5))['t'], names=('t',)); (t,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.derivation(), names=('X',)); (X,) = S._first_ngens(1)
>>> M = S.quotient_module((X + t)**Integer(2), names=('v', 'w',)); (v, w,) = M._first_ngens(2)
>>> Q = M.quo((X + t)*v)

>>> N = Q.relations()
>>> N
Ore module of rank 1 over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt
>>> (X + t)*v in N
True
>>> Q == M/N
True

Sage Live

K.<t> = Frac(GF(5)['t'])
S.<X> = OrePolynomialRing(K, K.derivation())
M.<v,w> = S.quotient_module((X + t)^2)
Q = M.quo((X + t)*v)
N = Q.relations()
N
(X + t)*v in N
Q == M/N

It is also possible to define names for the basis elements of $N$ :

Sage

sage: N.<u> = Q.relations()
sage: N
Ore module <u> over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt
sage: M(u)
v + 1/t*w

Python

>>> from sage.all import *
>>> N = Q.relations(names=('u',)); (u,) = N._first_ngens(1)
>>> N
Ore module <u> over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5 twisted by d/dt
>>> M(u)
v + 1/t*w

Sage Live

N.<u> = Q.relations()
N
M(u)

See also

relations()

rename_basis(names, coerce=False)[source]¶

Return the same Ore module with the given naming for the vectors in its distinguished basis.

INPUT:

names – a string or a list of strings, the new names
coerce (default: False) – a boolean; if True, a coercion map from this Ore module to the renamed version is set

EXAMPLES:

Sage

sage: K.<z> = GF(5^3)
sage: S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
sage: M = S.quotient_module(X^2 + z*X + 1)
sage: M
Ore module of rank 2 over Finite Field in z of size 5^3 twisted by z |--> z^5

sage: Me = M.rename_basis('e')
sage: Me
Ore module <e0, e1> over Finite Field in z of size 5^3 twisted by z |--> z^5

Python

>>> from sage.all import *
>>> K = GF(Integer(5)**Integer(3), names=('z',)); (z,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.frobenius_endomorphism(), names=('X',)); (X,) = S._first_ngens(1)
>>> M = S.quotient_module(X**Integer(2) + z*X + Integer(1))
>>> M
Ore module of rank 2 over Finite Field in z of size 5^3 twisted by z |--> z^5

>>> Me = M.rename_basis('e')
>>> Me
Ore module <e0, e1> over Finite Field in z of size 5^3 twisted by z |--> z^5

Sage Live

K.<z> = GF(5^3)
S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
M = S.quotient_module(X^2 + z*X + 1)
M
Me = M.rename_basis('e')
Me

Now compare how elements are displayed:

Sage

sage: M.random_element()   # random
(3*z^2 + 4*z + 2, 3*z^2 + z)
sage: Me.random_element()  # random
(2*z + 4)*e0 + (z^2 + 4*z + 4)*e1

Python

>>> from sage.all import *
>>> M.random_element()   # random
(3*z^2 + 4*z + 2, 3*z^2 + z)
>>> Me.random_element()  # random
(2*z + 4)*e0 + (z^2 + 4*z + 4)*e1

Sage Live

M.random_element()   # random
Me.random_element()  # random

At this point, there is no coercion map between M and Me. Therefore, adding elements in both parents results in an error:

Sage

sage: M.random_element() + Me.random_element()
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for +:
'Ore module of rank 2 over Finite Field in z of size 5^3 twisted by z |--> z^5' and
'Ore module <e0, e1> over Finite Field in z of size 5^3 twisted by z |--> z^5'

Python

>>> from sage.all import *
>>> M.random_element() + Me.random_element()
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for +:
'Ore module of rank 2 over Finite Field in z of size 5^3 twisted by z |--> z^5' and
'Ore module <e0, e1> over Finite Field in z of size 5^3 twisted by z |--> z^5'

Sage Live

M.random_element() + Me.random_element()

In order to set this coercion, one should define Me by passing the extra argument coerce=True:

Sage

sage: Me = M.rename_basis('e', coerce=True)
sage: M.random_element() + Me.random_element()  # random
2*z^2*e0 + (z^2 + z + 4)*e1

Python

>>> from sage.all import *
>>> Me = M.rename_basis('e', coerce=True)
>>> M.random_element() + Me.random_element()  # random
2*z^2*e0 + (z^2 + z + 4)*e1

Sage Live

Me = M.rename_basis('e', coerce=True)
M.random_element() + Me.random_element()  # random

Warning

Use coerce=True with extreme caution. Indeed, setting inappropriate coercion maps may result in a circular path in the coercion graph which, in turn, could eventually break the coercion system.

Note that the bracket construction also works:

Sage

sage: M.<v,w> = M.rename_basis()
sage: M
Ore module <v, w> over Finite Field in z of size 5^3 twisted by z |--> z^5

Python

>>> from sage.all import *
>>> M = M.rename_basis(names=('v', 'w',)); (v, w,) = M._first_ngens(2)
>>> M
Ore module <v, w> over Finite Field in z of size 5^3 twisted by z |--> z^5

Sage Live

M.<v,w> = M.rename_basis()
M

In this case, $v$ and $w$ are automatically defined:

Sage

sage: v + w
v + w

Python

>>> from sage.all import *
>>> v + w
v + w

Sage Live

v + w

class sage.modules.ore_module.OreSubmodule(ambient, basis, names)[source]¶

Bases: OreModule

Class for submodules of Ore modules.

ambient()[source]¶

Return the ambient Ore module in which this submodule lives.

EXAMPLES:

Sage

sage: K.<z> = GF(5^3)
sage: S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
sage: M.<v,w> = S.quotient_module((X + z)^2)
sage: N = M.span((X + z)*v)
sage: N.ambient()
Ore module <v, w> over Finite Field in z of size 5^3 twisted by z |--> z^5
sage: N.ambient() is M
True

Python

>>> from sage.all import *
>>> K = GF(Integer(5)**Integer(3), names=('z',)); (z,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.frobenius_endomorphism(), names=('X',)); (X,) = S._first_ngens(1)
>>> M = S.quotient_module((X + z)**Integer(2), names=('v', 'w',)); (v, w,) = M._first_ngens(2)
>>> N = M.span((X + z)*v)
>>> N.ambient()
Ore module <v, w> over Finite Field in z of size 5^3 twisted by z |--> z^5
>>> N.ambient() is M
True

Sage Live

K.<z> = GF(5^3)
S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
M.<v,w> = S.quotient_module((X + z)^2)
N = M.span((X + z)*v)
N.ambient()
N.ambient() is M

injection_morphism()[source]¶

Return the inclusion of this submodule in the ambient space.

EXAMPLES:

Sage

sage: K.<z> = GF(5^3)
sage: S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
sage: M.<v,w> = S.quotient_module((X + z)^2)
sage: N = M.span((X + z)*v)
sage: N.injection_morphism()
Ore module morphism:
  From: Ore module of rank 1 over Finite Field in z of size 5^3 twisted by z |--> z^5
  To:   Ore module <v, w> over Finite Field in z of size 5^3 twisted by z |--> z^5

Python

>>> from sage.all import *
>>> K = GF(Integer(5)**Integer(3), names=('z',)); (z,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.frobenius_endomorphism(), names=('X',)); (X,) = S._first_ngens(1)
>>> M = S.quotient_module((X + z)**Integer(2), names=('v', 'w',)); (v, w,) = M._first_ngens(2)
>>> N = M.span((X + z)*v)
>>> N.injection_morphism()
Ore module morphism:
  From: Ore module of rank 1 over Finite Field in z of size 5^3 twisted by z |--> z^5
  To:   Ore module <v, w> over Finite Field in z of size 5^3 twisted by z |--> z^5

Sage Live

K.<z> = GF(5^3)
S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
M.<v,w> = S.quotient_module((X + z)^2)
N = M.span((X + z)*v)
N.injection_morphism()

morphism_corestriction(f)[source]¶

If the image of $f$ is contained in this submodule, return the corresponding corestriction of $f$ .

EXAMPLES:

Sage

sage: K.<z> = GF(5^3)
sage: S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
sage: P = X + z
sage: M.<v,w> = S.quotient_module(P^2)
sage: N = M.span(P*v)

sage: f = M.hom({v: P*v})
sage: f
Ore module endomorphism of Ore module <v, w> over Finite Field in z of size 5^3 twisted by z |--> z^5

sage: g = N.morphism_corestriction(f)
sage: g
Ore module morphism:
  From: Ore module <v, w> over Finite Field in z of size 5^3 twisted by z |--> z^5
  To:   Ore module of rank 1 over Finite Field in z of size 5^3 twisted by z |--> z^5
sage: g.matrix()
[    z]
[4*z^2]

Python

>>> from sage.all import *
>>> K = GF(Integer(5)**Integer(3), names=('z',)); (z,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.frobenius_endomorphism(), names=('X',)); (X,) = S._first_ngens(1)
>>> P = X + z
>>> M = S.quotient_module(P**Integer(2), names=('v', 'w',)); (v, w,) = M._first_ngens(2)
>>> N = M.span(P*v)

>>> f = M.hom({v: P*v})
>>> f
Ore module endomorphism of Ore module <v, w> over Finite Field in z of size 5^3 twisted by z |--> z^5

>>> g = N.morphism_corestriction(f)
>>> g
Ore module morphism:
  From: Ore module <v, w> over Finite Field in z of size 5^3 twisted by z |--> z^5
  To:   Ore module of rank 1 over Finite Field in z of size 5^3 twisted by z |--> z^5
>>> g.matrix()
[    z]
[4*z^2]

Sage Live

K.<z> = GF(5^3)
S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
P = X + z
M.<v,w> = S.quotient_module(P^2)
N = M.span(P*v)
f = M.hom({v: P*v})
f
g = N.morphism_corestriction(f)
g
g.matrix()

When the image of the morphism is not contained in this submodule, an error is raised:

Sage

sage: h = M.multiplication_map(X^3)
sage: N.morphism_corestriction(h)
Traceback (most recent call last):
...
ValueError: the image of the morphism is not contained in this submodule

Python

>>> from sage.all import *
>>> h = M.multiplication_map(X**Integer(3))
>>> N.morphism_corestriction(h)
Traceback (most recent call last):
...
ValueError: the image of the morphism is not contained in this submodule

Sage Live

h = M.multiplication_map(X^3)
N.morphism_corestriction(h)

morphism_restriction(f)[source]¶

Return the restriction of $f$ to this submodule.

EXAMPLES:

Sage

sage: K.<z> = GF(5^3)
sage: S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
sage: M.<v,w> = S.quotient_module((X + z)^2)
sage: N = M.span((X + z)*v)

sage: f = M.multiplication_map(X^3)
sage: f
Ore module endomorphism of Ore module <v, w> over Finite Field in z of size 5^3 twisted by z |--> z^5

sage: g = N.morphism_restriction(f)
sage: g
Ore module morphism:
  From: Ore module of rank 1 over Finite Field in z of size 5^3 twisted by z |--> z^5
  To:   Ore module <v, w> over Finite Field in z of size 5^3 twisted by z |--> z^5
sage: g.matrix()
[        3 4*z^2 + 2]

Python

>>> from sage.all import *
>>> K = GF(Integer(5)**Integer(3), names=('z',)); (z,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.frobenius_endomorphism(), names=('X',)); (X,) = S._first_ngens(1)
>>> M = S.quotient_module((X + z)**Integer(2), names=('v', 'w',)); (v, w,) = M._first_ngens(2)
>>> N = M.span((X + z)*v)

>>> f = M.multiplication_map(X**Integer(3))
>>> f
Ore module endomorphism of Ore module <v, w> over Finite Field in z of size 5^3 twisted by z |--> z^5

>>> g = N.morphism_restriction(f)
>>> g
Ore module morphism:
  From: Ore module of rank 1 over Finite Field in z of size 5^3 twisted by z |--> z^5
  To:   Ore module <v, w> over Finite Field in z of size 5^3 twisted by z |--> z^5
>>> g.matrix()
[        3 4*z^2 + 2]

Sage Live

K.<z> = GF(5^3)
S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
M.<v,w> = S.quotient_module((X + z)^2)
N = M.span((X + z)*v)
f = M.multiplication_map(X^3)
f
g = N.morphism_restriction(f)
g
g.matrix()

rename_basis(names, coerce=False)[source]¶

Return the same Ore module with the given naming for the vectors in its distinguished basis.

INPUT:

names – a string or a list of strings, the new names
coerce (default: False) – a boolean; if True, a coercion map from this Ore module to renamed version is set

EXAMPLES:

Sage

sage: K.<z> = GF(5^3)
sage: S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
sage: M = S.quotient_module(X^2 + z^2)
sage: M
Ore module of rank 2 over Finite Field in z of size 5^3 twisted by z |--> z^5

sage: Me = M.rename_basis('e')
sage: Me
Ore module <e0, e1> over Finite Field in z of size 5^3 twisted by z |--> z^5

Python

>>> from sage.all import *
>>> K = GF(Integer(5)**Integer(3), names=('z',)); (z,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.frobenius_endomorphism(), names=('X',)); (X,) = S._first_ngens(1)
>>> M = S.quotient_module(X**Integer(2) + z**Integer(2))
>>> M
Ore module of rank 2 over Finite Field in z of size 5^3 twisted by z |--> z^5

>>> Me = M.rename_basis('e')
>>> Me
Ore module <e0, e1> over Finite Field in z of size 5^3 twisted by z |--> z^5

Sage Live

K.<z> = GF(5^3)
S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
M = S.quotient_module(X^2 + z^2)
M
Me = M.rename_basis('e')
Me

Now compare how elements are displayed:

Sage

sage: M.random_element()   # random
(3*z^2 + 4*z + 2, 3*z^2 + z)
sage: Me.random_element()  # random
(2*z + 4)*e0 + (z^2 + 4*z + 4)*e1

Python

>>> from sage.all import *
>>> M.random_element()   # random
(3*z^2 + 4*z + 2, 3*z^2 + z)
>>> Me.random_element()  # random
(2*z + 4)*e0 + (z^2 + 4*z + 4)*e1

Sage Live

M.random_element()   # random
Me.random_element()  # random

At this point, there is no coercion map between M and Me. Therefore, adding elements in both parents results in an error:

Sage

sage: M.random_element() + Me.random_element()
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for +:
'Ore module of rank 2 over Finite Field in z of size 5^3 twisted by z |--> z^5' and
'Ore module <e0, e1> over Finite Field in z of size 5^3 twisted by z |--> z^5'

Python

>>> from sage.all import *
>>> M.random_element() + Me.random_element()
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for +:
'Ore module of rank 2 over Finite Field in z of size 5^3 twisted by z |--> z^5' and
'Ore module <e0, e1> over Finite Field in z of size 5^3 twisted by z |--> z^5'

Sage Live

M.random_element() + Me.random_element()

In order to set this coercion, one should define Me by passing the extra argument coerce=True:

Sage

sage: Me = M.rename_basis('e', coerce=True)
sage: M.random_element() + Me.random_element()  # random
2*z^2*e0 + (z^2 + z + 4)*e1

Python

>>> from sage.all import *
>>> Me = M.rename_basis('e', coerce=True)
>>> M.random_element() + Me.random_element()  # random
2*z^2*e0 + (z^2 + z + 4)*e1

Sage Live

Me = M.rename_basis('e', coerce=True)
M.random_element() + Me.random_element()  # random

Warning

Use coerce=True with extreme caution. Indeed, setting inappropriate coercion maps may result in a circular path in the coercion graph which, in turn, could eventually break the coercion system.

Note that the bracket construction also works:

Sage

sage: M.<v,w> = M.rename_basis()
sage: M
Ore module <v, w> over Finite Field in z of size 5^3 twisted by z |--> z^5

Python

>>> from sage.all import *
>>> M = M.rename_basis(names=('v', 'w',)); (v, w,) = M._first_ngens(2)
>>> M
Ore module <v, w> over Finite Field in z of size 5^3 twisted by z |--> z^5

Sage Live

M.<v,w> = M.rename_basis()
M

In this case, $v$ and $w$ are automatically defined:

Sage

sage: v + w
v + w

Python

>>> from sage.all import *
>>> v + w
v + w

Sage Live

v + w

class sage.modules.ore_module.ScalarAction[source]¶

Bases: Action

Action by scalar multiplication on Ore modules.

sage.modules.ore_module.normalize_names(names, rank)[source]¶

Return a normalized form of names.

INPUT:

names – a string, a list of strings or None
rank – the number of names to normalize

EXAMPLES:

Sage

sage: from sage.modules.ore_module import normalize_names

Python

>>> from sage.all import *
>>> from sage.modules.ore_module import normalize_names

Sage Live

from sage.modules.ore_module import normalize_names

When names is a string, indices are added:

Sage

sage: normalize_names('e', 3)
('e0', 'e1', 'e2')

Python

>>> from sage.all import *
>>> normalize_names('e', Integer(3))
('e0', 'e1', 'e2')

Sage Live

normalize_names('e', 3)

When names is a list or a tuple, it remains untouched except that it is always casted to a tuple (in order to be hashable and serve as a key):

Sage

sage: normalize_names(['u', 'v', 'w'], 3)
('u', 'v', 'w')

Python

>>> from sage.all import *
>>> normalize_names(['u', 'v', 'w'], Integer(3))
('u', 'v', 'w')

Sage Live

normalize_names(['u', 'v', 'w'], 3)

Similarly, when names is None, nothing is returned:

Sage

sage: normalize_names(None, 3)

Python

>>> from sage.all import *
>>> normalize_names(None, Integer(3))

Sage Live

normalize_names(None, 3)

If the number of names is not equal to rank, an error is raised:

Sage

sage: normalize_names(['u', 'v', 'w'], 2)
Traceback (most recent call last):
...
ValueError: the number of given names does not match the rank of the Ore module

Python

>>> from sage.all import *
>>> normalize_names(['u', 'v', 'w'], Integer(2))
Traceback (most recent call last):
...
ValueError: the number of given names does not match the rank of the Ore module

Sage Live

normalize_names(['u', 'v', 'w'], 2)