Let R be a ring with unity e, R' a set, η a bijective map from R' into R. show that R' becomes a ring with unity if one defines:
a'+ b' = ηˉ¹ (η(a')+ η(b'))
a' b' = ηˉ¹(η(a') η(b'))
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.
Rings with Unity, Isomorphism, Bijectiveness and Invertibility are investigated.