When are the tensor product and the cartesian product of rings isomorphic?

Question

Let's say I'm interested in (commutative) unital rings, and I assume homomorphisms to preserve the unit. Are there interesting rings R and S such that $$R\otimes_\mathbb{Z}S\cong R\oplus S$$ holds? I know that $$R=S=0$$ works, but I'd be interested in seeing other examples or possibly proofs of those being impossible.

Answer 1

An example is $R = \mathbb{Z}$ and $S = \mathbb{Z}^\mathbb{N}$, the infinite product of countably many copies of $\mathbb{Z}$.

Answer 2

$$\mathbb{F}_{p^2}\otimes_\mathbb{Z}\mathbb{F}_{p^2}\cong\mathbb{F}_{p^2}\times\mathbb{F}_{p^2}$$

(Pretty much the same as the example over $\mathbb{R}$ that Qiaochu gave in his comment.)