Ring homomorphisms from a polynomial ring to another ring¶

This module currently implements the canonical ring homomorphism from $A [x]$ to $B [x]$ induced by a ring homomorphism from $A$ to $B$ .

Todo

Implement homomorphisms from $A [x]$ to an arbitrary ring $R$ , given by a ring homomorphism from $A$ to $R$ and the image of $x$ in $R$ .

AUTHORS:

Peter Bruin (March 2014): initial version

class sage.rings.polynomial.polynomial_ring_homomorphism.PolynomialRingHomomorphism_from_base[source]¶

Bases: RingHomomorphism_from_base

The canonical ring homomorphism from $R [x]$ to $S [x]$ induced by a ring homomorphism from $R$ to $S$ .

EXAMPLES:

sage: QQ['x'].coerce_map_from(ZZ['x'])
Ring morphism:
  From: Univariate Polynomial Ring in x over Integer Ring
  To:   Univariate Polynomial Ring in x over Rational Field
  Defn: Induced from base ring by
        Natural morphism:
          From: Integer Ring
          To:   Rational Field

is_injective()[source]¶

Return whether this morphism is injective.

EXAMPLES:

sage: R.<x> = ZZ[]
sage: S.<x> = QQ[]
sage: R.hom(S).is_injective()
True

is_surjective()[source]¶

Return whether this morphism is surjective.

EXAMPLES:

sage: R.<x> = ZZ[]
sage: S.<x> = Zmod(2)[]
sage: R.hom(S).is_surjective()
True