Modules of injective dimension one over Prüfer domains
Let \(R\) be an integral domain with quotient field \(Q\). \(R\) is called maximal (resp. almost maximal) if \(R/I\) is linearly compact for each ideal (resp. non-zero ideal) \(I\) of \(R\). According to \textit{I. Kaplansky} [Trans. Am. Math. Soc. 72, 327-340 (1952; Zbl 0046.25701)] and \textit{E. Matlis} [Nagoya Math. J. 15, 57-69 (1959; Zbl 0090.25305)], the following assertions are equivalent: (a) \(R\) is a maximal valuation domain, (b) every finite rank torsion-free \(R\)-module has injective dimension at most one and \(R\) is complete in the \(R\)-topology, (c) \(\text{Ext}_R^1(G,H)=0\) for all torsion-free \(R\)-modules such that \(H\) has finite rank, (d) \(\text{Ext}_R^1(X,Y)=0\) for all \(R\)-submodules \(X,Y\) of \(Q\), and (e) finite rank pure submodules of torsion-free \(R\)-modules are summands. In the paper under review, the author explores analogues for almost maximal Prüfer domains. Among many other results, he shows that \(R\) is an almost maximal Prüfer domain iff \(\text{Ext}_R^1(X,Y)\) is torsion-free for all \(R\)-submodules \(X,Y\) of \(Q\) such that \(X\subseteq Y\) iff each pure submodule of a submodule \(G\) of a finite rank free \(R\)-module is summand of \(G\). The technical kernel of the paper involves dominated torsion-free modules, the completion of an \(R\)-module in its \(R\)-topology and modules of injective dimension one.
scientific article; zbMATH DE number 1538670
scientific article