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# Hom(Module,Module) -- module of homomorphisms

## Synopsis

• Function: Hom
• Usage:
Hom(M,N)
• Inputs:
• M,
• N,
• Outputs:
• , The module Hom_R(M,N), where M and N are both R-modules

## Description

If M or N is an ideal or ring, it is regarded as a module in the evident way.

 i1 : R = QQ[x,y]/(y^2-x^3); i2 : M = image matrix{{x,y}} o2 = image | x y | 1 o2 : R-module, submodule of R i3 : H = Hom(M,M) o3 = image {-1} | x y | {-1} | y x2 | 2 o3 : R-module, submodule of R

To recover the modules used to create a Hom-module, use the function formation.

Specific homomorphisms may be obtained using homomorphism, as follows.

 i4 : f0 = homomorphism H_{0} o4 = {1} | 1 0 | {1} | 0 1 | o4 : Matrix i5 : f1 = homomorphism H_{1} o5 = {1} | 0 x | {1} | 1 0 | o5 : Matrix

In the example above, f0 is the identity map, and f1 maps x to y and y to x^2.

• homomorphism -- get the homomorphism from element of Hom
• Ext -- compute an Ext module
• compose -- composition as a pairing on Hom-modules
• formation -- recover the methods used to make a module

## Ways to use this method:

• "Hom(Ideal,Ideal)"
• "Hom(Ideal,Module)"
• "Hom(Ideal,Ring)"
• "Hom(Module,Ideal)"
• Hom(Module,Module) -- module of homomorphisms
• "Hom(Module,Ring)"
• "Hom(Ring,Ideal)"
• "Hom(Ring,Module)"