Ideals of injective dimension 1
type: Item
instance of: scholarly article
title: Ideals of injective dimension 1
zbMATH Open document ID: 0535.13011
author: Eben Matlis
published in: Michigan Mathematical Journal
publication date: 1982-01-01T00:00:00Z
review text: Let R be an integral domain with quotient field Q and let I be a non-zero ideal of R. In this paper the author investigates conditions under which the injective dimension of I is one (i.e. Q/I is injective). This leads naturally to the conditions \(E(R/I)=Q/I\) and \(E(R/I)\subseteq Q/I\) where E(R/I) denotes the injective envelope of R/I. The results relate these conditions to properties of certain overrings of R that the author calls complemented extensions, and to properties of torsion-free liftings and coverings. A ''complemented extension'' is defined as an R-submodule A of Q which contains R for which there exists an R-submodule A' of Q with \(A+A' = Q\) and \(A\cap A' = R\). --- These are investigated in section one where it is shown that A is a flat overring of R and A' is unique. It then follows that \(Q/I\cong Q/IA\oplus Q/IA'\) so these extensions can be expected to play a role in the above questions about Q/I. It is also shown that if H is the completion of R in the R-topology, then complemented extensions of R correspond to decompositions of H into a product of two subrings. -In section two it is shown that \(inj \dim_ RI=1\) is equivalent to the following equivalent conditions on I. (1) There exists a complemented extension A of R such that \(IA\cap R=I\), \(inj \dim_ AIA=1\), and \(inj \dim_{A'}A'\leq 1.\) (2) \(R_ S\) is a complemented extension of R, \(inj \dim_{R_ S}I_ S=1\) and \(inj \dim_{R'\!_ S}R'\!_ S\leq 1\) where \(S=\{1-a| a\in I\}\) and R'\({}_ S\) is the unique complement of \(R_ S\). A similar description of when \(E(R| I)\subseteq Q/I\) holds is also given. -In section three the author shows how the properties under consideration are related to torsion-free liftings and covers of various modules. For example this section contains the result that \(inj \dim_ RI=1\) is equivalent to each of the following conditions. \((1)\quad Ext^ 1\!_ R(X,IH)=0\) for any torsion-free R-module X. (2) The exact sequence \(0\to IH\to H\otimes Q\to Q/I\to 0…
Mathematics Subject Classification ID: 13C11
Mathematics Subject Classification ID: 13B02
Mathematics Subject Classification ID: 13G05
Mathematics Subject Classification ID: 13C05
Mathematics Subject Classification ID: 13B30
Mathematics Subject Classification ID: 13D05
Mathematics Subject Classification ID: 13F30
zbMATH DE Number: 3849380
zbMATH Keywords: injective dimension \(=\) 1
zbMATH Keywords: complemented extension
zbMATH Keywords: valuation rings
reviewed by: David E. Rush
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029002732
OpenAlex ID: W2003954130
Wikidata QID: Q114057222
DOI: 10.1307/MMJ/1029002732
description: scientific article; zbMATH DE number 3849380