Triangular matrix representations
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 definition, quasi-Baer if the right annihilator of each two-sided ideal of \(R\) is of the form \(eR\) for some (necessarily left semicentral) idempotent element \(e\) of \(R\). In particular a semicentral reduced quasi-Baer ring is prime. This makes it possible to give precise structural information about those quasi-Baer rings which have a complete GTMR, and then numerous known results follow as special cases including Gordon and Small's main theorem for piecewise domains.
generalized triangular matrix representations
scientific article; zbMATH DE number 1512980
scientific article