Explore BrainMass
Share

Explore BrainMass

    Rings and Modules : Quasi-Regular and Module Homomorphisms

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    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).

    Please see attached.

    © BrainMass Inc. brainmass.com October 9, 2019, 4:44 pm ad1c9bdddf
    https://brainmass.com/math/ring-theory/rings-modules-quasi-regular-module-homomorphisms-40192

    Attachments

    Solution Preview

    Please see the attached file for the full solution.
    Thanks for using BrainMass.

    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.

    $2.19