Let $R_1,R_2$ be two rings with identity. If for some $n\in\mathbb N$, $M_n(R_1)$ and $M_n(R_2)$ are isomorphic as rings, can we deduce that $R_1\cong R_2$? I can prove it when both $R_1,R_2$ are local or commutative.
2026-04-01 19:39:05.1775072345
Does $M_n(R_1)\cong M_n(R_2)$ imply $R_1\cong R_2$?
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