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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Vydáno v:	Linear algebra and its applications Ročník 434; číslo 4; s. 1018 - 1026
Hlavní autoři:	Ánh, P.N., van Wyk, L.
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Amsterdam Elsevier Inc 15.02.2011 Elsevier
Témata:	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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Shrnutí:	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