Explore BrainMass

Explore BrainMass

    Modules, Submodules, Commutative Rings with Unity and Homomorphism

    Not what you're looking for? Search our solutions OR ask your own Custom question.

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

    1. Let R and S be commutative rings with unity, and let φ: R S be a ring homomorphism. If M is an S-module, prove that M is also an R-module if we define rm = φ(r)m for all r E R and m E M.

    2. If M1 and M2 are submodules of an R-module M such that M = M1(+)M2, prove that M1 = M/M2 and M2 = M/M1.

    © BrainMass Inc. brainmass.com September 26, 2022, 9:11 pm ad1c9bdddf


    Solution Preview

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

    1. Proof:
    To show that is an -module, we have to verify the following conditions.
    (a) For any , , .
    Since is an -module, then we have

    (b) For any , ,
    Since is a ...

    Solution Summary

    Modules, Submodules , Commutative Rings with Unity and Homomorphism are investigated.