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Rings with Unity, Isomorphism, Bijectiveness and Invertibility

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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.

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

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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 ...

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