Explore BrainMass

Explore BrainMass

    Rings with Unity, Isomorphism, Bijectiveness and Invertibility

    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!

    Let R be a ring with unity e, R' a set, H a bijective map from R' into R. show that R' becomes a ring with unity if one defines:
    a'+ b' = Hˉ¹ (H(a')+ H(b'))
    a' b' = Hˉ¹(H(a') H(b'))
    0'= Hˉ¹(0)
    e'= Hˉ¹(e)
    and that is an isomorphism of R' with R. Use this to prove that if u is an invertible
    element of a ring then ( R, +, .u ,0,uˉ¹), where a .u b =aub is a ring is ring isomorphic to R.

    Show also that (R, θ ,o, 1, 0), where a θ b = a+b -1, a o b= a+b -ab is ring isomorphic to R.

    Please see attached for full question.

    © BrainMass Inc. brainmass.com December 24, 2021, 5:15 pm ad1c9bdddf


    Solution Preview

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

    First, I show that is a ring. For any , we know and . Thus . So is the additive identity of . Since and , then we have . So is the multiplicative identity. It is clear that is ...

    Solution Summary

    Rings with Unity, Isomorphism, Bijectiveness and Invertibility are investigated.