Answer to Question #17112 in Abstract Algebra for Tsit Lam

Question

Let R, S be rings such that Mm(R) ∼= Mn(S). Does this imply that m = n and R ∼ S?

Answer 1

The answer is “no”. For instance,for R = M2(Q) and S = M3(Q), we have R is not isomorphic to S (by dimension considerations), but M3(R) ∼= M2(S) since both rings are isomorphic to M6(Q).