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Rings with Unity, Isomorphism, Bijectiveness and Invertibility

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'))
0'= ηˉ¹(0)
e'= ηˉ¹(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.

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

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