# Ring / Ideal -- make a quotient ring

## Synopsis

• Usage:
S = R/I
• Operator: /
• Inputs:
• R, a ring
• I, an ideal or element of Ror a list or of elements of R
• Outputs:
• S, , the quotient ring R/I

## Description

If I is a ring element of R or ZZ, or a list or sequence of such elements, then I is understood to be the ideal generated by these elements. If I is a module, then it must be a submodule of a free module of rank 1.
 ```i1 : ZZ[x]/367236427846278621 ZZ[x] o1 = ------------------ 367236427846278621 o1 : QuotientRing```
 `i2 : A = QQ[u,v];` ```i3 : I = ideal random(A^1, A^{-2,-2,-2}) 9 2 1 9 2 1 2 3 2 3 2 3 7 2 o3 = ideal (-u + -u*v + -v , -u + u*v + -v , -u + -u*v + -v ) 2 2 4 2 4 2 4 4 o3 : Ideal of A``` `i4 : B = A/I;` `i5 : use A;` `i6 : C = A/(u^2-v^2,u*v);`
 ```i7 : D = GF(9,Variable=>a)[x,y]/(y^2 - x*(x-1)*(x-a)) o7 = D o7 : QuotientRing``` ```i8 : ambient D o8 = GF 9[x, y] o8 : PolynomialRing```
The names of the variables are assigned values in the new quotient ring (by automatically running use R) when the new ring is assigned to a global variable.

Warning: quotient rings are bulky objects, because they contain a Gröbner basis for their ideals, so only quotients of ZZ are remembered forever. Typically the ring created by R/I will be a brand new ring, and its elements will be incompatible with the elements of previously created quotient rings for the same ideal.

 ```i9 : ZZ/2 === ZZ/(4,6) o9 = true``` ```i10 : R = ZZ/101[t] o10 = R o10 : PolynomialRing``` ```i11 : R/t === R/t o11 = false```