Assume that the ring R is isomorphic to the ring R'. Prove that if R is commutative, then R' is commutative.

he298c 2021-01-27 Answered
Assume that the ring R is isomorphic to the ring R.
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Expert Answer

Delorenzoz
Answered 2021-01-28 Author has 91 answers

Definition of ring isomorphism:
Let R and R' denote two ring. A mapping ϕ:RR is a ring isomorphism from R to R' provided the following conditions hold:
1. ϕ is ont-to-ont correspondence (a bijection) from R to R'.
2. ϕ(x+y)=ϕ(x)+ϕ(y) for all xandyR.
3. ϕ(xy)=ϕ(X)ϕ(y) for all xandyR.
Proof:
Let R be an integral domain
Define a map ϕ:RR such that ϕ(x)=x
Let x,yR. Since pho is onto, there exists x,yR such that ϕ(x)=xandϕ(y)=y
Now xy=ϕ(x)ϕ(y)
Since ϕ is isomophism, from the definition of isomorphism,
=ϕ(xy)
Since R is a commutative ring,
=ϕ(yx)
=ρ(y)ϕ(x)
=yx
Hence if R is a commutative, then R' is commutative

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