# Singular Book 2.1.20 -- sum, intersection, module quotient

 i1 : A = QQ[x,y,z]; i2 : M = image matrix{{x*y,x},{x*z,x}} o2 = image | xy x | | xz x | 2 o2 : A-module, submodule of A i3 : N = image matrix{{y^2,x},{z^2,x}} o3 = image | y2 x | | z2 x | 2 o3 : A-module, submodule of A i4 : M + N o4 = image | xy x y2 x | | xz x z2 x | 2 o4 : A-module, submodule of A
Notice that, in Macaulay2, each module comes equipped with a list of generators, and operations such as sum do not try to simplify the list of generators.

Intersection, quotients, annihilators are found using standard notation:

 i5 : intersect(M,N) o5 = image | x xy2-xz2 | | x 0 | 2 o5 : A-module, submodule of A i6 : M : N o6 = ideal x o6 : Ideal of A i7 : N : M o7 = ideal(y + z) o7 : Ideal of A i8 : Q = A/x^5; i9 : M = substitute(M,Q) o9 = image | xy x | | xz x | 2 o9 : Q-module, submodule of Q i10 : ann M 4 o10 = ideal x o10 : Ideal of Q i11 : M : x o11 = image | 1 y-z x4 | | 1 0 0 | 2 o11 : Q-module, submodule of Q