# Module / Module -- quotient module

## Synopsis

• Operator: /
• Usage:
M/N
• Inputs:
• Outputs:
• , The quotient module M/N of M

## Description

If N is an ideal, ring element, or list or sequence of ring elements (in the ring of M), then the quotient is by the submodule N*M of M.

If N is a submodule of M, or a list or sequence of submodules, or a vector, then the quotient is by these elements or submodules.

 i1 : R = ZZ/173[a..d] o1 = R o1 : PolynomialRing i2 : M = ker matrix{{a^3-a*c*d,a*b*c-b^3,a*b*d-b*c^2}} o2 = image {3} | 0 b3-abc bc2-abd | {3} | -c2+ad a3-acd 0 | {3} | b2-ac 0 a3-acd | 3 o2 : R-module, submodule of R i3 : M/a == M/(a*M) o3 = true i4 : M/M_0 o4 = subquotient ({3} | 0 b3-abc bc2-abd |, {3} | 0 |) {3} | -c2+ad a3-acd 0 | {3} | -c2+ad | {3} | b2-ac 0 a3-acd | {3} | b2-ac | 3 o4 : R-module, subquotient of R i5 : M/(R*M_0 + b*M) o5 = subquotient ({3} | 0 b3-abc bc2-abd |, {3} | 0 0 b4-ab2c b2c2-ab2d |) {3} | -c2+ad a3-acd 0 | {3} | -c2+ad -bc2+abd a3b-abcd 0 | {3} | b2-ac 0 a3-acd | {3} | b2-ac b3-abc 0 a3b-abcd | 3 o5 : R-module, subquotient of R i6 : M/(M_0,a*M_1+M_2) o6 = subquotient ({3} | 0 b3-abc bc2-abd |, {3} | 0 ab3-a2bc+bc2-abd |) {3} | -c2+ad a3-acd 0 | {3} | -c2+ad a4-a2cd | {3} | b2-ac 0 a3-acd | {3} | b2-ac a3-acd | 3 o6 : R-module, subquotient of R i7 : presentation oo o7 = {5} | -1 0 -a3+acd | {6} | 0 -a -c2+ad | {6} | 0 -1 b2-ac | 3 3 o7 : Matrix R <--- R

## Ways to use this method:

• "Module / Ideal"
• "Module / List"
• Module / Module -- quotient module
• "Module / RingElement"
• "Module / Sequence"
• "Module / Vector"