Automorphism groups of generalized triangular matrix rings |
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Authors: | PN Ánh |
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Institution: | a Rényi Institute of Mathematics, Hungarian Academy of Sciences, 1364 Budapest, Pf. 127, Hungary b Department of Mathematical Sciences, Stellenbosch University, P/Bag X1, Matieland 7602, Stellenbosch, South Africa |
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Abstract: | We call a ring strongly indecomposable if it cannot be represented as a non-trivial (i.e. M≠0) generalized triangular matrix ring , 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 is unique up to isomorphism; to be more precise, if 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 |
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Keywords: | 16S50 15A33 16D20 |
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