# ideal(Ring) -- returns the defining ideal

## Synopsis

• Usage:
ideal R
• Function: ideal
• Inputs:
• Outputs:
• an ideal, which is the defining ideal of R

## Description

A quotient ring is a the quotient of its ambient ring by its defining ideal. Other rings have no ambient ring, and the defining ideal is its zero ideal.
 `i1 : S = ZZ/2[x,y,z];` ```i2 : ideal S o2 = ideal () o2 : Ideal of S``` ```i3 : R = S/(y^2-x*z,x^2*y-z^2) o3 = R o3 : QuotientRing``` ```i4 : ideal R 2 2 2 o4 = ideal (y + x*z, x y + z ) o4 : Ideal of S``` ```i5 : T = R/(x^3-y*z) o5 = T o5 : QuotientRing``` ```i6 : ideal T 3 o6 = ideal(x + y*z) o6 : Ideal of R``` ```i7 : ambient T o7 = R o7 : QuotientRing``` ```i8 : sing = singularLocus T o8 = sing o8 : QuotientRing``` ```i9 : ideal sing 3 2 2 2 2 3 4 2 2 o9 = ideal (x + y*z, y + x*z, x y + z , z , x + y*z, x*z, x , x y, x z, ------------------------------------------------------------------------ 3 x ) o9 : Ideal of S``` ```i10 : ambient sing o10 = S o10 : PolynomialRing```