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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Veröffentlicht in:Linear algebra and its applications Jg. 434; H. 4; S. 1018 - 1026
Hauptverfasser: Ánh, P.N., van Wyk, L.
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Amsterdam Elsevier Inc 15.02.2011
Elsevier
Schlagworte:
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
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Zusammenfassung: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