Module: sage.rings.quotient_ring
Author: William Stein
Module-level Functions
R, I) |
x) |
Class: QuotientRing_generic
sage: R = PolynomialRing(Z) sage: I = R.ideal([4 + 3*x + x^2, 1 + x^2]) sage: S = R.quotient_ring(I); S Quotient of Univariate Polynomial Ring in x over Integer Ring by the ideal (x^2 + 1, x^2 + 3*x + 4)
sage: R, (x,y) = PolynomialRing(Q, 2, 'xy').objgens() sage: S, (a,b) = (R/(x^2 + y^2)).objgens('ab') sage: a^2 + b^2 == 0 True sage: S(0) == a^2 + b^2 True
Quotient of quotient
A quotient of a quotient is just the quotient of the original top ring by the sum of two ideals.
sage: R, (x,y) = PolynomialRing(Q, 2, 'xy').objgens() sage: S, (a,b) = (R/(1 + y^2)).objgens('ab') sage: T, (c,d) = (S/(a, )).objgens('cd') sage: T Quotient of Polynomial Ring in x, y over Rational Field by the ideal (x, 1 + y^2) sage: T.gens() (0, d)
self, R, I, [names=None]) |
Create the quotient ring of R by the ideal I.
INPUT: R -- a commutative ring I -- an ideal
Functions: characteristic,
cover,
cover_ring,
defining_ideal,
gen,
is_field,
is_integral_domain,
lift,
ngens
self) |
The covering ring homomorphism
, equipped with a section.
sage: R = Z/(3*Z) sage: pi = R.cover() sage: pi Ring morphism: From: Integer Ring To: Ring of integers modulo 3 Defn: Natural quotient map sage: pi(5) 2 sage: sage: l = pi.lift()
sage: R, (x,y) = QQ['x,y'].objgens() sage: Q = R/(x^2,y^2) sage: pi = Q.cover() sage: pi(x^3+y) y sage: l = pi.lift(x+y^3) sage: l x sage: l = pi.lift(); l Set-theoretic ring morphism: From: Quotient of Polynomial Ring in x, y over Rational Field by the ideal (y^2, x^2) To: Polynomial Ring in x, y over Rational Field Defn: Choice of lifting map sage: l(x+y^3) x
self) |
Return the lifting map to the cover.
sage: R, (x,y) = PolynomialRing(Q, 2, 'xy').objgens() sage: S = R.quotient(x^2 + y^2, names=['xbar', 'ybar']) sage: pi = S.cover(); pi Ring morphism: From: Polynomial Ring in x, y over Rational Field To: Quotient of Polynomial Ring in x, y over Rational Field by the ideal (y^2 + x^2) Defn: Natural quotient map sage: L = S.lift(); L Set-theoretic ring morphism: From: Quotient of Polynomial Ring in x, y over Rational Field by the ideal (y^2 + x^2) To: Polynomial Ring in x, y over Rational Field Defn: Choice of lifting map sage: L(S.0) x sage: L(S.1) y
Note that some reduction may be applied so that the lift of a reduction need not equal the original element.
sage: z = pi(x^3 + 2*y^2); z 2*ybar^2 - xbar*ybar^2 sage: L(z) 2*y^2 - x*y^2 sage: L(z) == x^3 + 2*y^2 False
Special Functions: __call__,
__cmp__,
_latex_,
_repr_
self, other) |
Only quotients by the same (in "is") ring and same ideal are considered equal.