Automorphism groups of generalized triangular matrix rings

Abstract

We call a ring strongly indecomposable if it cannot be represented as a non-trivial (i.e. M ≠ 0 ) generalized triangular matrix ring R M 0 S , for some rings R and S and some R - S -bimodule R M S . 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 R M 0 S is unique up to isomorphism; to be more precise, if φ : R M 0 S → R ′ M ′ 0 S ′ 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 R M 0 S .