Isomorphisms between Morita context rings
Let (R, S, RM S, SN R,f,g) be a general Morita context, and let, be the ring associated with this context. Similarly, let, be another Morita context ring. We study the set Iso(T, T′) of ring isomorphisms from T to T′. Our interest in this problem is motivated by: (i) the problem to determine the automorphism group of the ring T, and (ii) the recovery of the non-diagonal tiles problem for this type of generalized matrix rings. We introduce two classes of isomorphisms from T to T′, the disjoint union of which is denoted by Iso 0(T, T′). We describe Iso 0(T, T′) by using the ℤ-graded ring structure of T and T′. Our main result characterizes Iso 0(T, T′) as the set consisting of all semigraded isomorphisms and all anti-semigraded isomorphisms from T to T′, provided that the rings R′ and S′ are indecomposable and at least one of M′ and N′ is nonzero; in particular, Iso 0(T, T′) contains all graded isomorphisms and all anti-graded isomorphisms from T to T′. We also present a situation where Iso 0(T, T′) = Iso(T, T′). This is in the case where R, S, R′ and S′ are rings having only trivial idempotents and all the Morita maps are zero. In particular, this shows that the group of automorphisms of T is completely determined. © 2012 Copyright Taylor and Francis Group, LLC.