Rings that are Morita equivalent to their opposites

We consider the following problem: Under what assumptions are one or more of the following 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. Based on the recent general bilinear forms of [10], we present a general machinery to attack the problem, and use it to show that (C)⟺(B)(C)⟺(B) when R   is semilocal or QQ-finite. Further results of similar flavor are also obtained, for example: If R   is a semilocal ring such that Mn(R)Mn(R) has an involution, then M2(R)M2(R) has an involution, and under further mild assumptions, R   itself has an involution. In contrast to that, we demonstrate that (B)⟹̸(A)(B)⟹̸(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.