Can $R \times R$ be isomorphic to $R$ as rings?

Question

I know from this question that $R \times R$ can be isomorphic to $R$, as $R$-modules.

But can they ever be isomorphic as rings?

Answer 1

Sure, if $R=\prod_{i\in\mathbb N} \mathbb Z$ then $R\times R\cong R$.

Answer 2

Sure. Take $R = \Pi_{i=1}^\infty \mathbb{Z}$ with the product ring structure. Then clearly $R \times R$ is isomorphic to $R$. To find an isomorphism, just use some bijection $\mathbb{N} \amalg \mathbb{N} \rightarrow \mathbb{N}$.