Can a commutative ring $R$ with identity be isomorphic to the product ring $R\times R$?

Question

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?

Answer 1

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?

Answer 2

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