MHB Is the direct sum of isomorphic commutative rings always equal in size?

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Here is this week's POTW:

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If $R$ is a nonzero commutative ring such that the direct sums $R^m$ and $R^n$ are isomorphic, show that $m = n$.

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No one answered this week’s problem. You can read my solution below.

Spoiler

Let $\mathfrak{m}$ be a maximal ideal of $R$, and consider the residue field $k := R/\mathfrak{m}$. If $f : R^m \to R^n$ is an isomorphism, then it induces an isomorphism $1 \otimes f : k \otimes_R R^m \to k \otimes_R R^n$. Now $k \otimes_R R^m$ and $k\otimes_R R^n$ are $k$-vector spaces of dimensions $m$ and $n$, respectively; since those vector spaces are isomorphic, $m = n$.