Triangular matrix representations
type: Item
instance of: scholarly article
title: Triangular matrix representations
zbMATH Open document ID: 0964.16031
author: G. F. Birkenmeier
author: Henry E. Heatherly
author: Jin Yong Kim
author: Jae Keol Park
published in: Journal of Algebra
publication date: 2001-07-03T00:00:00Z
review text: A ring \(R\) is said to have a GTMR (reviewer's abbreviation for ``generalised triangular matrix representation'') if there is a positive integer \(n\) such that \(R\) is isomorphic to the ring of all \(n\times n\) upper triangular matrices with \(i\)-th diagonal entry in some ring \(R_i\) and with above-diagonal \((i,j)\)-entry in an \(R_i\)-\(R_j\)-bimodule \(R_{ij}\). This is equivalent to \(R\) having a set of \(n\) non-zero idempotents \(b_1,\dots,b_n\) whose sum is \(1\) and such that \(b_iRb_j=0\) whenever \(i>j\). Such an ordered set \((b_1,\dots,b_n)\) is called a set of left triangulating idempotents for \(R\). Note that in these circumstances \(b_1R\) and \(Rb_n\) are two-sided ideals of \(R\). An idempotent element \(e\) of \(R\) is said to be left (resp. right) semicentral if \(eR\) (resp. \(Re\)) is a two-sided ideal of \(R\). Thus the existence of a GTMR for \(R\) is closely related to the presence of semicentral idempotents in \(R\) and certain of its subrings. We say that \(R\) has a complete GTMR if \(R\) has a GTMR in which each diagonal ring \(R_i\) is semicentral reduced (i.e., has no non-trivial semicentral idempotents). Thus the diagonal rings in a complete GTMR cannot be properly represented as triangular matrix rings. This paper carries out a detailed and systematic study of GTMR's for a ring \(R\). For instance if \(R\) has a complete GTMR in terms of \(n\times n\) triangular matrices then \(n\) is uniquely determined by \(R\) as also are, in an appropriate sense, the diagonal rings in the representation. If \(R\) has a complete GTMR then \(R\) has finite left global dimension if and only if each diagonal ring does, and bounds are given for these dimensions. Many classes of rings (e.g., rings with no infinite sets of orthogonal idempotents) have complete GTMR's and their study can be reduced in this way to that of semicentral reduced rings in the same class. Quasi-Baer rings are well-suited to this approach, because a ring \(R\) is, by defi…
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 16E10
Mathematics Subject Classification ID: 16D70
zbMATH DE Number: 1512980
zbMATH Keywords: generalized triangular matrix representations
zbMATH Keywords: rings of upper triangular matrices
zbMATH Keywords: left triangulating idempotents
zbMATH Keywords: semicentral idempotents
zbMATH Keywords: triangular matrix rings
zbMATH Keywords: left global dimensions
zbMATH Keywords: orthogonal idempotents
zbMATH Keywords: semicentral reduced rings
zbMATH Keywords: quasi-Baer rings
reviewed by: A. W. Chatters
MaRDI profile type: Publication
full work available at URL: jabr.2000.8328
OpenAlex ID: W2010230355
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DOI: 10.1006/JABR.2000.8328
description: scientific article; zbMATH DE number 1512980