Rings with isomorphic proper subrings

Rings will be unital here but I don't require that subrings share the identity elements with superrings.

I just accidentally came up with an example of a ring $R$ with a proper subring $S$ such that $R\cong S$ as rings. I may be mistaken, but I think this is the first such example I've ever come across. The example is this. $R$ is the ring of $\mathbb Z\times \mathbb Z$-matrices over $\mathbb R$ with finite-support columns, and $S$ is the subring of matrices with zeroes in all places indexed by at least one negative number. Then $S$ is isomorphic to the ring of $\mathbb N\times\mathbb N$-matrices over $\mathbb R$. But then $R$ is isomorphic to the endomorphism ring of the direct sum of $\aleph_0$ copies of $\mathbb R$ and $S$ is too.

I would like to know if there are other examples of such rings and if it has anything to do with the rings being Dedekind-infinite. I suspect it might because Dedekind-infinite sets are those which are equinumerous with some of their proper subsets.

Even simpler $R[X]\simeq R[X^n] \subset R[X]$.
More simply, let $R$ be any ring with more than one element and let $R^{\mathbb{N}}$ be the set of all sequences of elements of $R$, made into a ring by the obvious pointwise operations. Then the subset of $R^{\mathbb{N}}$ consisting of sequences with first term $0$ is a proper subring isomorphic to $R^{\mathbb{N}}$ itself.