Torsion-free Modules Research Articles

Torsion-free, Divisible, and Mittag-Leffler Modules

We study (relative) 𝒦-Mittag–Leffler modules, with emphasis on the class 𝒦 of absolutely pure modules. A final goal is to describe the 𝒦-Mittag–Leffler abelian groups as those that are, modulo their torsion part, ℵ1-free. Several more general results of independent interest are derived on the way. In particular, every flat 𝒦-Mittag–Leffler module (for 𝒦 as before) is Mittag–Leffler. A question about the definable subcategories generated by the divisible modules and the torsion-free modules, resp., has been left open.

Separable torsion-free modules with UA-rings of endomorphisms

In this paper we study a property of unique addition for an endomorphisms ring of torsion-free separable module over commutative Dedekind ring.

A fast flatness testing criterion in characteristic zero

We prove a fast computable criterion that expresses non-flatness in terms of torsion: Let R be a regular algebra of finite type over a field K of characteristic zero and let F be a module finitely generated over an R-algebra of finite type. Given a maximal ideal m in R, let S be the coordinate ring of the blowing-up of Spec(R) at the closed point m. Then F is flat over R localized in m if and only if the tensor product of F with S over R is a torsion-free module over R localized in m. If K is the field of reals or complex numbers, we give a stronger criterion - without the regularity assumption on R. We also show the corresponding results in the real- and complex-analytic categories.

A semigroup-theoretical view of direct-sum decompositions and associated combinatorial problems

Let R be a ring and let [Formula: see text] be a small class of right R-modules which is closed under finite direct sums, direct summands, and isomorphisms. Let [Formula: see text] denote a set of representatives of isomorphism classes in [Formula: see text] and, for any module M in [Formula: see text], let [M] denote the unique element in [Formula: see text] isomorphic to M. Then [Formula: see text] is a reduced commutative semigroup with operation defined by [M] + [N] = [M ⊕ N], and this semigroup carries all information about direct-sum decompositions of modules in [Formula: see text]. This semigroup-theoretical point of view has been prevalent in the theory of direct-sum decompositions since it was shown that if EndR(M) is semilocal for all [Formula: see text], then [Formula: see text] is a Krull monoid. Suppose that the monoid [Formula: see text] is Krull with a finitely generated class group (for example, when [Formula: see text] is the class of finitely generated torsion-free modules and R is a one-dimensional reduced Noetherian local ring). In this case, we study the arithmetic of [Formula: see text] using new methods from zero-sum theory. Furthermore, based on module-theoretic work of Lam, Levy, Robson, and others we study the algebraic and arithmetic structure of the monoid [Formula: see text] for certain classes of modules over Prüfer rings and hereditary Noetherian prime rings.

Monoids of modules and arithmetic of direct-sum decompositions

Let $R$ be a (possibly noncommutative) ring and let $\mathcal C$ be a class of finitely generated (right) $R$-modules which is closed under finite direct sums, direct summands, and isomorphisms. Then the set $\mathcal V (\mathcal C)$ of isomorphism classes of modules is a commutative semigroup with operation induced by the direct sum. This semigroup encodes all possible information about direct sum decompositions of modules in $\mathcal C$. If the endomorphism ring of each module in $\mathcal C$ is semilocal, then $\mathcal V (\mathcal C)$ is a Krull monoid. Although this fact was observed nearly a decade ago, the focus of study thus far has been on ring- and module-theoretic conditions enforcing that $\mathcal V(\mathcal C)$ is Krull. If $\mathcal V(\mathcal C)$ is Krull, its arithmetic depends only on the class group of $\mathcal V(\mathcal C)$ and the set of classes containing prime divisors. In this paper we provide the first systematic treatment to study the direct-sum decompositions of modules using methods from Factorization Theory of Krull monoids. We do this when $\mathcal C$ is the class of finitely generated torsion-free modules over certain one- and two-dimensional commutative Noetherian local rings.

ON LCM-STABLE MODULES

A torsion-free module M over a commutative integral domain R is said to be LCM-stable over R if (Ra ∩ Rb)M = Ma ∩ Mb for all a, b ∈ R. We show that if the module M is LCM-stable over a GCD-domain R, then the polynomial module M[X] is LCM-stable over R[X]; if R is a w-coherent locally GCD-domain, then LCM-stability and reflexivity are equivalent for w-finite type torsion-free R-modules. Finally, we introduce the concept of w-LCM-stability for modules over a domain. Then we characterize when the module M is w-LCM-stable over the domain in terms of localizations and t-Nagata modules, respectively. Also we characterize Prüfer v-multiplication domains and Krull domains in terms of w-LCM-stability.

Harmonic Maps and Von Staudt’s Theorem Over Rings

We use U. Brehm’s ideas to study cross-ratios, harmonic relations, and harmonic maps. For torsion-free modules over noncommutative principal ideal domains, von Staudt–Hua’s theorem is proved. Moreover, more general (nonbijective) harmonic maps with the classical definition of harmonic quadruples are calculated.

N-almost prime submodules

Let R be a commutative ring with identity. A proper submodule N of an R-module M will be called prime [resp. n-almost prime], if for r ∈ R and a ∈ M with ra ∈ N [resp. ra ∈ N \ (N: M) n−1 N], either a ∈ N or r ∈ (N: M). In this note we will study the relations between prime, primary and n-almost prime submodules. Among other results it is proved that: (1) If N is an n-almost prime submodule of an R-module M, then N is prime or N = (N: M)N, in case M is finitely generated semisimple, or M is torsion-free with dim R = 1. (2) Every n-almost prime submodule of a torsion-free Noetherian module is primary. (3) Every n-almost prime submodule of a finitely generated torsion-free module over a Dedekind domain is prime. (4) There exists a finitely generated faithful R-module M such that every proper submodule of M is n-almost prime, if and only if R is Von Neumann regular or R is a local ring with the maximal ideal m such that m 2 = 0. (5) If I is an n-almost prime ideal of R and F is a flat R-module with IF ≠ F, then IF is an n-almost prime submodule of F.

Krull Modules

In this article, we generalize the concepts of several classes of domains (which are related to Krull domains) to torsion-free modules, and show that for a faithful multiplication module M over an integral domain R, M is a Krull module if and only if R is a Krull domain. Then we characterize Krull, Dedekind, and factorial modules.

Pure projective torsion free modules over Bass domains

We prove that every pure projective torsion free module over a (commutative) Bass domain with finite normalization contains a finitely generated direct summand.

Continuous closure of sheaves

We give a purely algebraic construction of the continuous closure of any finitely generated torsion free module; a concept first studied by H.~Brenner and M.~Hochster. The construction implies that, at least in characteristic 0, taking continuous closure commutes with flat morphisms whose fibers are semi-normal. This implies that the continuous closure of a coherent ideal sheaf is again a coherent ideal sheaf (both in the Zariski and in the etale topologies) and it commutes with field extensions.

On the Unions of Ascending Chains of Direct Sums of Ideals of h-Local Prüfer Domains

In this work, we investigate conditions under which unions of ascending chains of modules which are isomorphic to direct sums of ideals of an integral domain are again isomorphic to direct sums of ideals. We obtain generalizations of the Pontryagin-Hill theorems for modules which are direct sums of ideals of h-local Prüfer domains. Particularly, we prove that a torsion-free module over a Dedekind domain with a countable number of maximal ideals is isomorphic to a direct sum of ideals if it is the union of a countable ascending chain of pure submodules which are isomorphic to direct sums of ideals.

Monoids of torsion-free modules over rings with finite representation type

Monoids of torsion-free modules over rings with finite representation type

On the Union of Increasing Chains of Torsion-Free Modules Over Integral Domains

Motivated by Hill’s criterion of freeness for abelian groups, we establish a generalization of that result to categories \({\mathcal{C}}\) of torsion-free modules over integral domains, which are closed with respect to the formation of direct sums, and in which every object can be decomposed into direct sums of objects of \({\mathcal{C}}\) of rank at most a fixed limit cardinal number κ. Our main result states that a module belongs to \({\mathcal{C}}\) if it is the union of a continuous, well-ordered, ascending chain of length κ, consisting of pure submodules which are objects of \({\mathcal{C}}\) . As corollaries, we derive versions of Hill’s theorem for some classes of torsion-free modules over domains, and a generalization of a well-known result by Kaplansky.

Torsion in equivariant cohomology and Cohen-Macaulay G-actions

We show that the well-known fact that the equivariant cohomology (with real coefficients) of a torus action is a torsion-free module if and only if the map induced by the inclusion of the fixed point set is injective generalises to actions of arbitrary compact connected Lie groups if one replaces the fixed point set by the set of points with isotropy rank equal to the rank of the acting group. This is true essentially because the action on this set is always equivariantly formal. In case this set is empty we show that the induced action on the set of points with highest occuring isotropy rank is Cohen-Macaulay. It turns out that just as equivariant formality of an action is equivalent to equivariant formality of the action of a maximal torus, the same holds true for equivariant injectivity and the Cohen-Macaulay property. In addition, we find a topological criterion for equivariant injectivity in terms of orbit spaces.

On completely decomposable and separable modules over Prüfer domains

We generalize known results on summands of completely decomposable and separable torsion-free abelian groups to modules over h-local Pr\"ufer domains. Over such domains summands of completely decomposable torsion-free modules are again completely decomposable (Theorem 3.2) and summands of separable torsion-free modules are likewise separable (Theorem 4.2). In addition, a Pontryagin-Hill type theorem is established on countable chains of homogeneous completely decomposable modules over h-local Pr\"ufer domains.

Large tilting modules and representation type

We study finiteness conditions on large tilting modules over arbitrary rings. We then turn to a hereditary artin algebra R and apply our results to the (infinite dimensional) tilting module L that generates all modules without preprojective direct summands. We show that the behaviour of L over its endomorphism ring determines the representation type of R. A similar result holds true for the (infinite dimensional) tilting module W that generates the divisible modules. Finally, we extend to the wild case some results on Baer modules and torsion-free modules proven in Angeleri Hugel, L., Herbera, D., Trlifaj, J.: Baer and Mittag-Leffler modules over tame hereditary algebras. Math. Z. 265, 1–19 (2010) for tame hereditary algebras.

Star operations on modules

We present a theory of (semi)star operations for torsion-free modules. This extends the analogous theory of star operations on domains as in [R. Gilmer, Multiplicative Ideal Theory, M. Dekker, New York, 1972] and its generalization to semistar operations studied in [A. Okabe, R. Matsuda, Semistar operations on integral domains, Math. J. Toyama Univ. 17 (1994) 1–21], and recovers some closure operations defined on modules. We investigate some properties of (semi)star operations on a given module over a domain D and their relation with the properties of some classes of semistar operations induced on D. Among other things, this leads to a connection between semistar operations on the D-module and localizing systems on the domain D.

Bounds for indecomposable torsion-free modules

Let M be a finitely generated torsion-free module over a one-dimensional reduced Noetherian ring R with finitely generated normalization. The rank of M is the tuple of vector-space dimensions of M P over each field R P ( R localized at P ), where P ranges over the minimal prime ideals of R . We assume that there exists a bound N R on the ranks of all indecomposable finitely generated torsion-free R -modules. For such rings, what bounds and ranks occur? Partial answers to this question have been given by a plethora of authors over the past forty years. In this article we provide a final answer by giving a concise list of the ranks of indecomposable modules for R a local ring with no condition on the characteristic. We conclude that if the rank of an indecomposable module M is ( r , r , … , r ) , then r ∈ { 1 , 2 , 3 , 4 , 6 } , even when R is not local.

Notes on Divisible and Torsionfree Modules

In this article, we first study the existence of envelopes and covers by modules of finite divisible and torsionfree dimensions. Then we investigate divisible and torsionfree dimensions as well as localizations of divisible and torsionfree modules over commutative rings. Finally, Gorenstein divisible and torsionfree modules are introduced and studied.