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This is a simple question, but I have no idea with this. Can a commutative ring $R$ with identity be isomorphic to the product ring $R\times R$?

It is clear that this is not true when $R$ is an integral domain, because $R$ has no zero divisors while $(0,1)(1,0)=0$ in $R\times R$. But does this still false in the general case?

blancket
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  • Please search for your questions first. This one was at the top of the list of related questions on the right hand side column. – rschwieb Apr 03 '20 at 17:54

2 Answers2

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Just take a non-zero ring $S$, and consider the infinite Cartesian product $$R=S\times S\times S\times\cdots.$$

Now, are there any Noetherian examples?

Angina Seng
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There are counterexamples, the easiest being from to the observation that $R^{\Bbb N}\times R^{\Bbb N}\cong R^{\Bbb N}$