11.9 Quotient Ring Elements

Module: sage.rings.quotient_ring_element

Author: William Stein

Class: QuotientRingElement

class QuotientRingElement
An element of a quotient ring $ R/I$ .

sage: R, x = PolynomialRing(Z).objgen()
sage: S = R/(4 + 3*x + x^2, 1 + x^2); S
Quotient of Univariate Polynomial Ring in x over Integer Ring by the ideal
(x^2 + 1, x^2 + 3*x + 4)
sage: v = S.gens(); v
(x,)

sage: loads(v[0].dumps()) == v[0]  # todo: not implemented
True

sage: R, (x,y) = PolynomialRing(Z, 2, 'xy').objgens()
sage: S = R/(x^2 + y^2); S
Quotient of Polynomial Ring in x, y over Integer Ring by the ideal (y^2 +
x^2)
sage: S.gens()
(x, y)

We name each of the generators.

sage: S, (a,b) = (R/(x^2 + y^2)).objgens('ab')
sage: a
a
sage: b
b
sage: b.lift()
y
sage: (a^3 + b^2).lift()
y^2 + x^3
QuotientRingElement( self, parent, rep, [reduce=True])

Functions: copy,$  $ is_unit,$  $ is_zero,$  $ lift

Special Functions: __float__,$  $ __int__,$  $ __invert__,$  $ __long__,$  $ __neg__,$  $ __pos__,$  $ __pow__,$  $ __rdiv__,$  $ _add_,$  $ _cmp_,$  $ _div_,$  $ _integer_,$  $ _mul_,$  $ _rational_,$  $ _reduce_,$  $ _repr_,$  $ _sub_

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