In this MO answer it is stated that every finite ring is a direct sum of finite-dimensional algebras over $\mathbb{Z}/p^k$ for varying $p$ and $k.$ What I am wondering is the following:
Can every finite ring $R$ be written as a subring of $\text{Mat}_{n\times n}A$ for some commutative ring $A$?
If not, then what is/are the smallest ring(s) that cannot be?
Sorry if this question is trivial for some reason (something like the regular representation or permuation representation?), or if I should have asked a question about algebras, or ideals, or modules, instead.
Apparently the answer is no, and the smallest counterexample has order $2^5 = 32$. I suspect it is $\text{End}(\mathbb{Z}_2 \oplus \mathbb{Z}_4)$, but don't quote me on that. (By looking at the action of a finite ring on itself by left multiplication you can reduce the question to endomorphism rings of finite abelian groups, and by looking at each prime separately you can reduce the question to endomorphism rings of finite abelian $p$-groups. That ring above is the simplest such ring which isn't already a matrix ring.)