# Singular Book 2.1.10 -- Submodules of A^n

A common method of creating a submodule of A^n in Macaulay2 is to take the image of a matrix. This will be a submodule generated by the columns of the matrix.
 i1 : A = QQ[x,y,z]; i2 : f = matrix{{x*y-1,y^4},{z^2+3,x^3},{x*y*z,z^2}} o2 = | xy-1 y4 | | z2+3 x3 | | xyz z2 | 3 2 o2 : Matrix A <--- A i3 : M = image f o3 = image | xy-1 y4 | | z2+3 x3 | | xyz z2 | 3 o3 : A-module, submodule of A i4 : numgens M o4 = 2 i5 : ambient M 3 o5 = A o5 : A-module, free
A submodule can easily be moved to quotient rings.
 i6 : Q = A/(x^2+y^2+z^2); i7 : substitute(M,Q) o7 = image | xy-1 y4 | | z2+3 -xy2-xz2 | | xyz z2 | 3 o7 : Q-module, submodule of Q