Automorphism groups of generalized triangular matrix rings

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 endomorphi...

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Bibliographic Details
Published in:Linear algebra and its applications Vol. 434; no. 4; pp. 1018 - 1026
Main Authors: Ánh, P.N., van Wyk, L.
Format: Journal Article
Language:English
Published: Amsterdam Elsevier Inc 15.02.2011
Elsevier
Subjects:
Algebra
Associative rings and algebras
Automorphism group
Bimodule isomorphism
Exact sciences and technology
Linear and multilinear algebra, matrix theory
Mathematics
Sciences and techniques of general use
Strongly indecomposable ring
Upper triangular matrix ring
16D20
Strongly indecomposable ring
16S50
15A33
Upper triangular matrix ring
Automorphism group
Bimodule isomorphism
Triangular matrix
Triangulation
Idempotent matrix
Upper triangular matrix
Bimodule
Vector space
Endomorphism
Ring
Isomorphism
ISSN:0024-3795
Online Access:Get full text
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Summary: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 .
ISSN:0024-3795
DOI:10.1016/j.laa.2010.10.007