Rings That Are Morita Equivalent to Their Opposites

Abstract

We consider the following problem: Under what assumptions do one or more of the following are equivalent for a ring $R$: (A) $R$ is Morita equivalent to a ring with involution, (B) $R$ is Morita equivalent to a ring with an anti-automorphism, (C) $R$ is Morita equivalent to its opposite ring. The problem is motivated by a theorem of Saltman which roughly states that all conditions are equivalent for Azumaya algebras. Basing on the recent "general bilinear forms", we present a general machinery to attack the problem, and use it to show that (C)$\iff$(B) when $R$ is semilocal or $\mathbb{Q}$-finite. Further results of similar flavor are also obtained, for example: If $R$ is a semilocal ring such that $\mathrm{M}_{n}(R)$ has an involution, then $\mathrm{M}_{2}(R)$ has an involution, and under further mild assumptions, $R$ itself has an involution. In contrast to that, we demonstrate that (B) does not imply (A). Our methods also give a new perspective on the Knus-Parimala-Srinivas proof of Saltman's Theorem. Finally, we give a method to test Azumaya algebras of exponent $2$ for the existence of involutions, and use it to construct explicit examples of such algebras.