On inert modules over valuation domains
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On inert modules over valuation domains Luigi Salce1 Published online: 24 June 2019 © Akadémiai Kiadó, Budapest, Hungary 2019
Abstract We start the investigation of inert modules over valuation domains, a class of modules containing finitely generated and quasi-injective modules. A complete description is provided when the valuation domain is a DVR. For arbitrary valuation domains, we reduce the investigation to reduced torsion modules and obtain a complete characterization of inert uniserial modules. Keywords Valuation domains · Finitely generated modules · Quasi-injective modules · Inert modules · Uniserial modules Mathematics Subject Classification Primary: 13F30 · 13C05; Secondary: 13C12 · 13C13
1 Introduction Two most investigated classes of modules over arbitrary rings are the class of finitely generated modules and the class of quasi-injective modules, characterized by the property of being fully invariant in their injective hull. A class of modules which provides a common generalization of these two classes consists of inert modules. A left R-module M over a unitary ring R is called inert if it is fully inert in its injective hull E(M), that is, for every endomorphism φ of E(M) the quotient module (M + φ M)/M is finitely generated. Evidently, finitely generated and quasi-injective modules are inert. Notice that, when R = Z, the condition that (M + φ M)/M is finitely generated can be replaced by saying that (M + φ M)/M is finite, because E(M)/M is a torsion group and finitely generated torsion groups are finite. It is worthwhile to recall that inert Abelian groups arose as a subproduct of the investigation of dynamical properties of endomorphisms of Abelian groups, using the tool of the intrinsic algebraic entropy (see [3]). The structure of inert Abelian groups is completely described in [2, Theorem 5.5]. Furthermore, fully inert subgroups of some classes of Abelian groups and of J p -modules have been investigated in [4,7,8]. If R is an integral domain and M is an R-module, then E(M)/M is a torsion module, so in the investigation of inert R-modules the structure of finitely generated torsion R-modules
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Luigi Salce [email protected] Dipartimento di Matematica “Tullio Levi-Civita”, Università di Padova, Via Trieste 63, 35121 Padua, Italy
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On inert modules over valuation domains
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plays a central role. These modules are well understood in case R is a valuation domain (see [5,6]). Also quasi-injective modules over these domains are well known (see [5, Chapter VI.6]); actually, they have been recently classified by cardinal invariants in [12]. Motivated by these facts, in this paper we start the investigation of inert modules over valuation domains. In Sect. 2 we adapt some results obtained in [2] on inert Abelian groups to inert modules over a discrete rank one valuation domain, providing their complete characterization. In Sect. 3 we investigate inert modules over arbitrary valuation domains R. As the structure of injective modules is well known (see [16] or [6, Chapter IX.4]), we concentrate
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