Torsion-free Modules Research Articles

Modules with reduced endomorphism rings

In this paper, we study endo-reduced modules as modules whose endomorphism rings have no nonzero nilpotent elements. We characterize their properties for different classes of modules, including [Formula: see text]-non-singular modules, multiplication modules and finitely generated modules over commutative Dedekind domains. In the subcategory of finitely generated modules, it is shown that the class of rings [Formula: see text] for which every faithful multiplication [Formula: see text]-module is endo-reduced is precisely that of reduced rings; while the class of rings [Formula: see text] for which every multiplication [Formula: see text]-module is endo-reduced is precisely that of von Neumann regular rings. Characterizations of when an endo-reduced module will be a reduced module are given. We prove that a finitely generated module over a principal ideal domain (PID) is endo-reduced exactly if it is either a semisimple module with pair-wise non-isomorphic submodules or a torsion-free module which is isomorphic to the underlying ring.

Big pure projective modules over commutative noetherian rings: Comparison with the completion

Abstract A module over a ring R is pure projective provided it is isomorphic to a direct summand of a direct sum of finitely presented modules. We develop tools for the classification of pure projective modules over commutative noetherian rings. In particular, for a fixed finitely presented module M, we consider Add ⁡ ( M ) {\operatorname{Add}(M)} , which consists of direct summands of direct sums of copies of M. We are primarily interested in the case where R is a one-dimensional, local domain, and in torsion-free (or Cohen–Macaulay) modules. We show that, even in this case, Add ⁡ ( M ) {\operatorname{Add}(M)} can have an abundance of modules that are not direct sums of finitely generated ones. Our work is based on the fact that such infinitely generated direct summands are all determined by finitely generated data. Namely, idempotent/trace ideals of the endomorphism ring of M and finitely generated projective modules modulo such idempotent ideals. This allows us to extend the classical theory developed to study the behaviour of direct sum decomposition of finitely generated modules comparing with their completion to the infinitely generated case. We study the structure of the monoid V * ⁢ ( M ) {V^{*}(M)} , of isomorphism classes of countably generated modules in Add ⁡ ( M ) {\operatorname{Add}(M)} with the addition induced by the direct sum. We show that V * ⁢ ( M ) {V^{*}(M)} is a submonoid of V * ⁢ ( M ⊗ R R ^ ) {V^{*}(M\otimes_{R}\widehat{R})} , this allows us to make computations with examples and to prove some realization results.

The finitely presented torsion-free SG-projective modules are not necessarily projective

It is proved in [Strongly Gorenstein projective, injective and flat modules, J. Algebra 320 (2008) 2659–2674, Theorem 2.8] that a finitely presented torsion-free module is SG-projective if and only if it is projective. Let L⊆F be an extension of fields with [F:L]=2. Set R=L+XF[X], m=XF[X] and D=R[Y]. Via the study of w-invertibility over infra-Krull domains, we prove that the w-conductor m[Y] of D is SG-projective but not projective, and all maximal w-ideals other than m[Y] are w-invertible. It follows that a finitely presented torsion-free SG-projective module do not need to be projective.

$\boldsymbol{R}$-regular modules

The notion of $R$-regular module is introduced. We will show that if $M$ is a torsion-free module over a commutative ring, then $M$ is $R$-regular module if and only if $M$ is a strongly regular module. We will also show that if $M$ is a cyclic $R$-regular $IFP$ module, then the submodule $P$ is completely prime if and only if $P$ is maximal.

The construction D + XDS [X] and G-Prüfer domains

Let D be an integral domain with quotient field K, S a (not necessarily saturated) multiplicative subset of D and . In this paper, we show that is a G-Prüfer domain if and only if D is a G-Prüfer domain and DS = K. We also prove that if D is a Bass domain (or a local NWF domain), then every finitely generated torsion-free module over is isomorphic to a direct sum of some ideals.

A Schreier-type property for modules

We extend to torsion-free modules over integral domains the theory of (pre)-Schreier domains initiated by Cohn and Zafrullah.

TORSION-FREE EXTENSIONS OF PROJECTIVE MODULES BY TORSION MODULES

We consider a generalization of a problem raised by P. Griffith on abelian groups to modules over integral domains, and prove an analogue of a theorem of M. Dugas and J. Irwin. Torsion modules T with the following property are characterized: if M is a torsion-free module and F is a projective submodule such that M∕F≅⁡T, then M is projective. It is shown that for abelian groups whose cardinality is not cofinal with ω this is equivalent to being totally reduced in the sense of L. Fuchs and K. Rangaswamy. The problem for valuation domains is also discussed, with results similar to the case of abelian groups.

On subflat domains of RD-flat modules

The concept of subflat domain is used to measure how close (or far away) a module is to be flat. A right module is flat if its subflat domain is the entire class of left modules. In this note, we focus on of RD-flat modules that have subflat domain which is exactly the collection of all torsion-free modules, shortly tf-test modules. Properties of subflat domains and of tf-test modules are studied. New characterizations of left P-coherent rings and torsion-free rings by subflat domains of cyclically presented left $R$-modules are obtained.

Polynomial extensions of Krull modules

Let M be a torsion-free module over an integral domain D with quotient field K. In this paper, we show that M is a Krull module over D in the sense of Costa-Johnson (Inert extensions of Krull domains, 1976) if and only if the polynomial module is a Krull module over in the sense of Costa-Johnson.

Irreducibility criteria for factorial modules

Let M be a torsion-free module over an integral domain R. The main results of this article provide analogue of Dumas irreducibility criterion and an extension of Eisenstein’s irreducibility criterion for factorial modules. Furthermore, we provide some applications of these results in -module .

Generalized lattices over one-dimensional noetherian domains

We study direct sum decompositions of pure projective torsion free modules over one-dimensional commutative noetherian domains. Drawing inspiration from the representation theory of orders in separable algebras, we study when every pure projective torsion free module is a direct sum of finitely generated modules. A satisfactory criterion is given for analytically unramified reduced local rings and for Bass domains.

On normal modules

Recall that a commutative ring is said to be a normal ring if it is reduced and every two distinct minimal prime ideals are comaximal. A finitely generated reduced R-module is said to be a normal module if every two distinct minimal prime submodules are comaximal. The concepts of normal modules and locally torsion free modules are different, whereas they are equal in theory of commutative rings. We give many properties and examples of normal modules, we use them to characterize locally torsion free modules and Baer modules. Also, we give the topological characterizations of normal modules.

Gorenstein‐projective modules over short local algebras

Following the well-established terminology in commutative algebra, any (not necessarily commutative) finite-dimensional local algebra A $A$ with radical J $J$ will be said to be short provided J 3 = 0 $J^3 = 0$ . As in the commutative case, we show: If a short local algebra A $A$ has an indecomposable non-projective Gorenstein-projective module M $M$ , then either A $A$ is self-injective (so that all modules are Gorenstein-projective) and then, of course, | J 2 | ⩽ 1 $|J^2| \leqslant 1$ , or else | J 2 | = | J / J 2 | − 1 $|J^2| = |J/J^2| - 1$ and | J M | = | J 2 | | M / J M | $|JM| = |J^2||M/JM|$ . More generally, we focus the attention to semi-Gorenstein-projective and ∞ $\infty$ -torsionfree modules, even to ℧ $\mho$ -paths of length 2, 3 and 4. In particular, we show that the existence of a non-projective reflexive module implies that | J 2 | < | J / J 2 | $|J^2| < |J/J^2|$ and further restrictions. In addition, we consider exact complexes of projective modules with a non-projective image. Again, as in the commutative case, we see that if such a complex exists, then A $A$ is self-injective or satisfies the condition | J 2 | = | J / J 2 | − 1 $|J^2| = |J/J^2| - 1$ . Also, we show that any non-projective semi-Gorenstein-projective module M $M$ satisfies Ext 1 ( M , M ) ≠ 0 $\operatorname{Ext}^1(M,M) \ne 0$ . In this way, we prove the Auslander-Reiten conjecture (one of the classical homological conjectures) for arbitrary short local algebras. Many arguments used in the commutative case actually work in general, but there are interesting differences and some of our results may be new also in the commutative case.

On Unique Factorization Modules: A Submodule Approach

Let M be a torsion-free module over an integral domain D. We define a concept of a unique factorization module in terms of v-submodules of M. If M is a unique factorization module (UFM), then D is a unique factorization domain. However, the converse situation is not necessarily to be held, and we give four different characterizations of unique factorization modules. Further, it is shown that the concept of the UFM is equivalent to Nicolas’s UFM, which is defined in terms of irreducible elements of D and M.

The Decomposition of a Finitely Generated Module over Some Special Ring

This research aims to give the decompositions of a finitely generated module over some special ring, such as the principal ideal domain and Dedekind domain. One of the main problems with module theory is to analyze the objects of the module. This research was using a literature study on finitely generated modules topics from scientific journals, especially those related to the module theory. And by selective cases we find a pattern to build a conjecture or a hypothesis, by deductive proof, we prove the conjecture and state it as a theorem. The main result in this study is the decomposition of the finitely generated module is a direct sum of the torsion submodule and torsion-free submodule. Since the torsion-free module is always a free module over a principal ideal domain, then the torsion-free submodule is a free module. Last, we generalize the ring, from a principal ideal domain, to a Dedekind domain. We found then the torsion-free submodule became a projective module. Then the decomposition of the finitely generated module is a direct sum of the torsion submodule and the projective submodule. These results should help the researchers to analyze the objects on module theory.

RD-projective module whose subprojectivity domain is minimal

A p-indigent module is one that is subprojective only to projective modules. An RD-projective module is subprojective to any torsionfree (and flat) module. An RD-projective module $T$ is called rdp-indigent if it is subprojective only to torsionfree modules. In this work, we consider the structure of SRDP rings whose (simple) RD-projective right $R$-modules are rdp-indigent or torsionfree. Moreover, new characterizations of P-coherent rings and torsionfree rings are presented by subprojectivity domains.

Locally torsion-free modules

Recall that a commutative ring [Formula: see text] is a locally integral domain if its localization [Formula: see text] is an integral domain for each prime ideal [Formula: see text] of [Formula: see text] Our aim in this paper is to extend the notion of locally integral domains to modules. Let [Formula: see text] be a commutative ring with a unity and [Formula: see text] a nonzero unital [Formula: see text]-module. [Formula: see text] is called a locally torsion-free module if the localization [Formula: see text] of [Formula: see text] is a torsion-free [Formula: see text]-module for each prime ideal [Formula: see text] of [Formula: see text] In addition to giving many properties of locally torsion-free modules, we use them to characterize Baer modules, torsion free modules, and von Neumann regular rings.

On $S$-comultiplication modules

Let $R\ $be a commutative ring with $1\neq0$ and $M$ be an $R$-module. Suppose that $S\subseteq R\ $is a multiplicatively closed set of $R.\ $Recently Sevim et al. in \cite{SenArTeKo} introduced the notion of an $S$-prime submodule which is a generalization of a prime submodule and used them to characterize certain classes of rings/modules such as prime submodules, simple modules, torsion free modules,\ $S$-Noetherian modules and etc. Afterwards, in \cite{AnArTeKo}, Anderson et al. defined the concepts of $S$-multiplication modules and $S$-cyclic modules which are $S$-versions of multiplication and cyclic modules and extended many results on multiplication and cyclic modules to $S$-multiplication and $S$-cyclic modules. Here, in this article, we introduce and study $S$-comultiplication modules which are the dual notion of $S$-multiplication module. We also characterize certain classes of rings/modules such as comultiplication modules, $S$-second submodules, $S$-prime ideals and $S$-cyclic modules in terms of $S$-comultiplication modules. Moreover, we prove $S$-version of the dual Nakayama's Lemma.

Krull modules and completely integrally closed modules

Let [Formula: see text] be a torsion-free module over an integral domain [Formula: see text] with quotient field [Formula: see text]. We define a concept of completely integrally closed modules in order to study Krull modules. It is shown that a Krull module [Formula: see text] is a [Formula: see text]-multiplication module if and only if [Formula: see text] is a maximal [Formula: see text]-submodule and [Formula: see text] for every minimal prime ideal [Formula: see text] of [Formula: see text]. If [Formula: see text] is a finitely generated Krull module, then [Formula: see text] is a Krull module and [Formula: see text]-multiplication module. It is also shown that the following three conditions are equivalent: [Formula: see text] is completely integrally closed, [Formula: see text] is completely integrally closed, and [Formula: see text] is completely integrally closed.

Residual intersections and modules with Cohen-Macaulay Rees algebra

In this paper, we consider a finite, torsion-free module E over a Gorenstein local ring. We provide sufficient conditions for E to be of linear type and for the Rees algebra R(E) of E to be Cohen-Macaulay. Our results are obtained by constructing a generic Bourbaki ideal I of E and exploiting properties of the residual intersections of I.