# Rings and Modules : Quasi-Regular and Module Homomorphisms

1. Let R be a ring. Prove that if x, y E R such that xy is right quasi-regular then yx is also right quasi-regular.

3. Let M and N be left R-modules. Let f : M N and g : N M be left R-module homomorphisms such that fg(y) = y for all y N. Show that M = ker(f) im(g).

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1. Proof:

We know, is right quasi-regular in a ring if and only if is right invertible. i.e. there exists some , such that .

Now for any , if is right quasi-regular, then there exists , such that ...

#### Solution Summary

Quasi-regular elements of a ring, module homomorphisms and kernels are investigated. The solution is detailed and well presented. The response was given a rating of "5" by the student who originally posted the question.