Outer Automorphism Group of Matrix Rings

According to the theorem of Isaacs, the outer automorphism group of the matrix algebra $ M_{n}(R) $ , where $ R $ is a unique factorization domain, is trivial for every $ n\in{𝕅} $ . We study generalizations of this theorem. It is proved that the outer automorphism group of the matrix algebra over an arbitrary highest common factor domain is trivial. For the algebra of formal matrices $ {𝕄}_{n}(R;s) $ over a unique factorization domain $ R $ , the outer automorphism group is determined. As a consequence, we obtain a criterion for the isomorphism between the algebra $ {𝕄}_{n}(R;s) $ and the algebra of formal matrices of order $ n $ with entries in $ R $ .