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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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.