Automorphism groups of generalized triangular matrix rings

AbstractWe call a ring strongly indecomposable if it cannot be represented as a non-trivial (i.e. M≠0) generalized triangular matrix ring RM0S, for some rings R and S and some R-S-bimodule RMS. Examples of such rings include rings with only the trivial idempotents 0 and 1, as well as endomorphism rings of vector spaces, or more generally, semiprime indecomposable rings. We show that if R and S are strongly indecomposable rings, then the triangulation of the non-trivial generalized triangular matrix ring RM0S is unique up to isomorphism; to be more precise, if φ:RM0S→R′M′0S′ is an isomorphism, then there are isomorphisms ρ:R→R′ and ψ:S→S′ such that χ:=φ∣M:M→M′ is an R-S-bimodule isomorphism relative to ρ and ψ. In particular, this result describes the automorphism groups of such upper triangular matrix rings RM0S