A ring isomorphic to its square Is there an example of a ring $A$ (with unity) which is isomorphic as unital rings to $A\times A$?
Any such ring can't have invariant basis number so in particular can't be commutative. 

$R^{\omega}\cong R^{\omega}\times R^{\omega}$ for any ring $R$, commutative or not. This does not contradict the fact that a commutative ring always has the invariant basis number property because while we can construct an "artificial" isomorphism of rings, it is not an isomorphism of $A$-modules, as the action of $A$ is different. So $A\times A$ really is a free module of rank $2$, and if $A$ is commutative then it is not a free $A$-module of rank $1$.
