# submodules and quotients

## submodules

We can create submodules by using standard mathematical notation, keeping in mind that the generators of a module M are denoted by M_0, M_1, etc.
 i1 : R = QQ[x,y,z]; i2 : M = R^3 3 o2 = R o2 : R-module, free i3 : I = ideal(x^2,y^2-x*z) 2 2 o3 = ideal (x , y - x*z) o3 : Ideal of R
Here are some examples of submodules of M.
 i4 : I*M o4 = image | x2 0 0 y2-xz 0 0 | | 0 x2 0 0 y2-xz 0 | | 0 0 x2 0 0 y2-xz | 3 o4 : R-module, submodule of R i5 : R*M_0 o5 = image | 1 | | 0 | | 0 | 3 o5 : R-module, submodule of R i6 : I*M_1 o6 = image | 0 0 | | x2 y2-xz | | 0 0 | 3 o6 : R-module, submodule of R i7 : J = I*M_1 + R*y^5*M_1 + R*M_2 o7 = image | 0 0 0 0 | | x2 y2-xz y5 0 | | 0 0 0 1 | 3 o7 : R-module, submodule of R
To determine if one submodule is contained in the other, use isSubset(Module,Module).
 i8 : isSubset(I*M,M) o8 = true i9 : isSubset((x^3-x)*M,x*M) o9 = true
Another way to construct submodules is to take the kernel or image of a matrix.
 i10 : F = matrix{{x,y,z}} o10 = | x y z | 1 3 o10 : Matrix R <--- R i11 : image F o11 = image | x y z | 1 o11 : R-module, submodule of R i12 : kernel F o12 = image {1} | -y 0 -z | {1} | x -z 0 | {1} | 0 y x | 3 o12 : R-module, submodule of R
The module M does not need to be a free module. We will see examples below.

## quotients

If N is a submodule of M, construct the quotient using Module / Module.
 i13 : F = R^3 3 o13 = R o13 : R-module, free i14 : F/(x*F+y*F+R*F_2) o14 = cokernel | x 0 0 y 0 0 0 | | 0 x 0 0 y 0 0 | | 0 0 x 0 0 y 1 | 3 o14 : R-module, quotient of R
When constructing M/N, it is not necessary that M be a free module, or a quotient of a free module. In this case, we obtain a subquotient module, which we describe below.