# Ideal / Ideal -- quotient module

## Synopsis

• Operator: /
• Usage:
I/J
• Inputs:
• Outputs:
• , The quotient module (I+J)/J

## Description

 i1 : R = QQ[a,b,c] o1 = R o1 : PolynomialRing i2 : I = ideal vars R o2 = ideal (a, b, c) o2 : Ideal of R i3 : M = I / I^2 o3 = subquotient (| a b c |, | a2 ab ac b2 bc c2 |) 1 o3 : R-module, subquotient of R
There is a diffference between typing I/J and (I+J)/J in Macaulay2, although conceptually they are the same module. The former has as its generating set the generators of I, while the latter has as its (redundant) generators the generators of I and J. Generally, the former method is preferable.
 i4 : gens M o4 = | a b c | 1 3 o4 : Matrix R <--- R i5 : N = (I + I^2)/I^2 o5 = subquotient (| a b c a2 ab ac b2 bc c2 |, | a2 ab ac b2 bc c2 |) 1 o5 : R-module, subquotient of R i6 : gens N o6 = | a b c a2 ab ac b2 bc c2 | 1 9 o6 : Matrix R <--- R