Gorenstein triangular matrix rings and category algebras》.pdfVIP

Journal of Pure and Applied Algebra 220 (2016) 666–682 Contents lists available at ScienceDirect Journal of Pure and Applied Algebra /locate/jpaa Gorenstein triangular matrix rings and category algebras Ren Wang School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, PR China a r t i c l e i n f o a b s t r a c t Article history: We give conditions on when a triangular matrix ring is Gorenstein of a given Received 21 January 2015 selfinjective dimension. We apply the result to the category algebra of a finite EI Received in revised form 22 June category. In particular, we prove that for a finite EI category, its category algebra 2015 is 1-Gorenstein if and only if the given category is free and projective. Available online 17 August 2015 © 2015 Elsevier B.V. All rights reserved. Communicated by S. Iyengar MSC: Primary: 16G10; secondary: 16D90; 18E30 1. Introduction Let R be a ring with a unit. Recall that R is Gorenstein if R is two-sided noetherian satisfying idRR ∞ and idRR ∞. Here, we use id to denote the injective dimension of a module. It is well known that for a Gorenstein ring R we have idRR = idRR ; see [15, Lemma A]. Let m ≥ 0. A Gorenstein ring R is m-Gorenstein if idRR = idRR ≤ m. We observe that a 0-Gorenstein ring coincides with a quasi-Frobenius ring. ⎛R1 M12 · · · M1n ⎞ ⎜ R ·