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¶
Bases:
sage.rings.morphism.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()¶
Return whether this morphism is injective.
EXAMPLES:
sage: R.<x> = ZZ[] sage: S.<x> = QQ[] sage: R.hom(S).is_injective() True
- is_surjective()¶
Return whether this morphism is surjective.
EXAMPLES:
sage: R.<x> = ZZ[] sage: S.<x> = Zmod(2)[] sage: R.hom(S).is_surjective() True