The flat dimensions of injective modules
type: Item
instance of: scholarly article
title: The flat dimensions of injective modules
zbMATH Open document ID: 0804.16005
DOI: 10.1007/BF02599307
published in: Manuscripta Mathematica
publication date: 1995-01-12T00:00:00Z
full work available at URL: https://eudml.org/doc/155802
review text: If \(M\) is a module, let \(\text{fd}(M)\), \(\text{pd}(M)\) and \(\text{id}(M)\) denote the flat dimension, projective dimension, and injective dimension of \(M\), respectively. Let \(R\) be an associative ring with identity and \(\text{Mod-}R\) (\(R\text{-Mod})\) be the category of unitary right (left) \(R\)-modules. The authors prove, among other things, that for any ring \(R\) we have, \(l.\text{FWD}(R) \leq r.\text{IFD}(R) \leq \text{WD}(R)\), where \(\text{WD}(R)\) is the weak dimension of the ring \(R\), \(l.\text{FWD}(R) = \sup\{\text{fd}(M) \mid M \in R\text{-Mod}\) with \(\text{fd}(M) < \infty\}\) and \(r.\text{IFD}(R) = \sup\{\text{fd}(E)\mid E\in \text{Mod- }R\) with \(E\) injective\}. Let \(M^ + = \text{Hom}_ \mathbb{Z}(M,\mathbb{Q}/\mathbb{Z})\) for an \(R\)-module. The authors also prove that the following are identical for any ring \(R\): (1) \(r.\text{IFD}(R)\), (2) \(\sup\{\text{fd}(M) \mid M \in \text{Mod-}R\) with \(M\) FP-injective\}, (3) \(\sup\{\text{fd}(M) \mid M \in \text{Mod-}R\) with \(\text{FP-id}(M) < \infty\}\), and (4) \(\text{sup}\{\text{fd}(M^ +) \mid M \in R\text{- Mod}\) with \(\text{fd}(M) < \infty\}\).
Mathematics Subject Classification ID: 16E10
Mathematics Subject Classification ID: 16D50
zbMATH DE Number: 492770
zbMATH Keywords: flat dimension
zbMATH Keywords: projective dimension
zbMATH Keywords: injective dimension
zbMATH Keywords: weak dimension
author: Nanqing Ding
author: Jianlong Chen
reviewed by: Xue Weimin
MaRDI profile type: Publication
Wikidata QID: Q114233773
cites work: Q4056148
cites work: Finitistic Dimension and a Homological Generalization of Semi-Primary Rings
cites work: Injective Dimension in Noetherian Rings
cites work: Flat and Projective Character Modules
cites work: On the free product of associative rings
cites work: Rings which have flat injective modules
cites work: Homological properties of the ring of differential polynomials
cites work: Balanced functors applied to modules
cites work: Q4745957
cites work: Character modules
cites work: Character modules, dimension and purity
cites work: On rings with finite self-injective dimension. II
cites work: Flat and FP-Injectivity
cites work: On the vanishing of \(\lim_{(i)}\)
cites work: Global dimension of fully bounded noetherian rings
cites work: Q3882615
cites work: Coherent Rings and Fp -Injective Modules
cites work: Injective dimension of semi-primary rings
OpenAlex ID: W1987677846
description: scientific article; zbMATH DE number 492770