Finitistic dimensions of serial-like rings
type: Item
instance of: scholarly article
title: Finitistic dimensions of serial-like rings
zbMATH Open document ID: 0821.16007
DOI: 10.1016/0022-4049(94)90089-2
author: Yong Wang
published in: Journal of Pure and Applied Algebra
publication date: 1995-09-28T00:00:00Z
review text: Let \(R\) be a left Artinian ring, then the finitistic dimensions of \(R\) are defined as follows: \(\text{lfPD}(R)= \sup\{\text{Pd}(M)\mid M\) a finitely generated left \(R\)-module of \(\text{Pd}(M)< \infty\}\), \(\text{lFPD}(R) =\sup\{\text{Pd}(M)\mid M\) a left \(R\)-module of \(\text{Pd}(M)< \infty\}\), where \(\text{Pd}(M)\) denotes the projective dimension of \(M\). The author determines bounds for the finitistic dimensions of one-sided almost serial rings in terms of the projective dimensions of the simple modules of finite projective dimension and in terms of the number of the simple modules. More precisely, he shows that: 1) For a left almost serial ring \(R\), \(\text{lFPD}(R)\leq\text{Pd}(Rf/Jf)+ 1\), where \(f\) is an idempotent such that the left simple modules appearing in \(Rf/Jf\) are exactly the ones having finite projective dimensions; 2) For a two-sided Artinian left or right almost serial ring, \(\text{lFPD}(R)\leq 2n- 2\), where \(n\) is the number of simple left \(R\)-modules. He also proves that, for an Artinian left uniformly monomial ring \(R\), \(\text{lFPD}(R)= \text{lfPD}(R)\).
reviewed by: Nguyen Viet Dung
Mathematics Subject Classification ID: 16E10
Mathematics Subject Classification ID: 16P20
Mathematics Subject Classification ID: 16D60
zbMATH DE Number: 681828
zbMATH Keywords: left Artinian rings
zbMATH Keywords: finitistic dimension
zbMATH Keywords: finitely generated left \(R\)- modules
zbMATH Keywords: projective dimension
zbMATH Keywords: almost serial rings
zbMATH Keywords: simple modules
MaRDI profile type: Publication
full work available at URL: 0022-4049(94)90089-2
OpenAlex ID: W2011501221
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description: scientific article; zbMATH DE number 681828