up previous next
InducedHom

homomorphism induced by a homomorphism

Syntax
InducedHom(RmodI: RING, phi: RINGHOM): RINGHOM

Description
InducedHom(RmodI, phi) -- where RmodI is a QuotientRing, and phi is a homomorphism R --> S (which must have BaseRing(RmodI) as its domain , and whose ker must contain DefiningIdeal(RmodI) ) gives the homomorphism R/I --> S induced by phi

InducedHom(FrF, phi) -- may be partial where FrF is a FractionField, gives the homomorphism induced by phi (which must have the base ring of FrF as its domain). Note that the resulting homomorphism may be only partial (e.g. if ker(phi) is non-trivial, or if the codomain is not a field).

Example
/**/  use R ::= QQ[x,y];
/**/  RmodI := NewQuotientRing(R, ideal(x^2-1));

/**/ use S ::= QQ[a,b,c];
/**/  SmodJ := NewQuotientRing(S, ideal(a^2-1));

/**/  phi := PolyAlgebraHom(R,S,[a,b]);
/**/  use R;
/**/  phi(x);
a
/**/ RingOf(phi(x)) = S;
true
/**/  psi := CanonicalHom(S,SmodJ)(phi);  -- composition of homomorphisms
/**/  psi(x);
(a)
/**/  RingOf(psi(x)) = SmodJ;
true

/**/  theta := InducedHom(RmodI, psi);
/**/  use RmodI;
/**/  theta(x);
(a)

See Also