Rings with Unity, Isomorphism, Bijectiveness and Invertibility
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 October 9, 2019, 4:38 pm ad1c9bdddfhttps://brainmass.com/math/ring-theory/rings-unity-isomorphism-bijectiveness-invertibility-37500
Attachments
Solution Preview
Please see the attached file for the complete solution.
Thanks for using BrainMass.
Proof:
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.