Torsion-free R-modules Research Articles

A Note On E-Injective Modules

Let 𝑅 be a commutative ring with identity, 𝑀 an 𝑅-module and 𝐸 a torsion-free 𝑅-module. A 
 submodule 𝑁 of 𝑀 is said to be essential (large) in 𝑀 if the intersection of 𝑁 with each nonzero 
 submodule of 𝑀 is nonzero, that is, 𝑁 ∩ 𝑅𝑚 ≠ 0 for any nonzero element 𝑚 ∈ 𝑀 and we write 
 𝑁 ≤𝑒 𝑀. It is clear that the class of 𝑒 − 𝑒𝑥𝑎𝑐𝑡 sequences is larger than the class of 𝑒𝑥𝑎𝑐𝑡
 sequences. In this study we present the concept of e-injective modules as a generalization of 
 injective modules. The main goal is to give a characterization of e-injective modules in terms of 
 contravariant functor 𝐻𝑜𝑚(−,𝐸)

On w∞ -projective modules and Krull domains

Let R be a commutative ring with identity. In this paper, w∞ -projective modules are introduced and studied. It is shown that every R-module has a special w∞ -projective precover. As an application, it is proved that a domain R is a Krull domain if and only if every submodule of a w∞ -projective R-module is w∞ -projective. And we show that for any Krull domain R with where denotes the class of all strong w-modules and denotes the class of -torsionfree R-modules N with the property that for all w-projective R-modules M and all integers

Domination number of total graphs

Abstract Let R be a commutative ring with Z(R) the set of zero-divisors and U(R) the set of unit elements of R. The total graph of R, denoted by T(Γ(R)), is the (undirected) graph with all elements of R as vertices, and for distinct x, y ∈ R, the vertices x and y are adjacent if and only if x + y ∈ Z(R). We study the domination number of T(Γ(R)). It is shown that if R = Z(R) ∪ U(R), then the domination number of T(∪(R)) is finite provided R has a maximal ideal of finite index. Moreover, if R = ∏ i = 1 n F i $R = \prod\limits_{i = 1}^n {{F_i}} $ , where Fi is a field for each 1 ≤ i ≤ n and t = |F 1| ≤ |F 2| ≤ ··· ≤ |Fn |, then the domination number of T(Γ(R)) is equal to t - 1 provided t = |Fi | for every 1 ≤ i ≤ n, and is equal to t otherwise. Finally, for an R-module M it is shown that the total domination number of the total graph of the idealization (Nagata extension) R(+)M is equal to the domination number of the total graph of R provided M is a torsion free R-module or R = Z(R) ∪ U(R).

Idempotent generated algebras and Boolean powers of commutative rings

A Boolean power S of a commutative ring R has the structure of a commutative R-algebra, and with respect to this structure, each element of S can be written uniquely as an R-linear combination of orthogonal idempotents so that the sum of the idempotents is 1 and their coefficients are distinct. In order to formalize this decomposition property, we introduce the concept of a Specker R-algebra, and we prove that the Boolean powers of R are up to isomorphism precisely the Specker Ralgebras. We also show that these algebras are characterized in terms of a functorial construction having roots in the work of Bergman and Rota. When R is indecomposable, we prove that S is a Specker R-algebra iff S is a projective R-module, thus strengthening a theorem of Bergman, and when R is a domain, we show that S is a Specker R-algebra iff S is a torsion-free R-module. For indecomposable R, we prove that the category of Specker R-algebras is equivalent to the category of Boolean algebras, and hence is dually equivalent to the category of Stone spaces. In addition, when R is a domain, we show that the category of Baer Specker R-algebras is equivalent to the category of complete Boolean algebras, and hence is dually equivalent to the category of extremally disconnected compact Hausdorff spaces. For totally ordered R, we prove that there is a unique partial order on a Specker R-algebra S for which it is an f-algebra over R, and show that S is isomorphic to the R-algebra of piecewise constant continuous functions from a Stone space X to R equipped with the interval topology.

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.

Flatness testing over singular bases

We show that non-flatness of a morphism f of complex-analytic spaces with a locally irreducible target Y of dimension n manifests in the existence of vertical components in the n-fold fibred power of the pull-back of f to the desingularization of Y. An algebraic analogue follows: Let R be a locally (analytically) irreducible finite type complex-algebra and an integral domain of Krull dimension n, and let S be a regular n-dimensional algebra of finite type over R (but not necessarily a finite R-module), such that the induced morphism of spectra is dominant. Then a finite type R-algebra A is R-flat if and only if the tensor product of S with the n-fold tensor power of A over R is a torsion-free R-module.

ω-MODULES OVER COMMUTATIVE RINGS

Let R be a commutative ring and let M be a GV -torsionfree R-module. Then M is said to be a <TEX>$\omega$</TEX>-module if <TEX>$Ext_R^1$</TEX>(R/J, M) = 0 for any J <TEX>$\in$</TEX> GV (R), and the w-envelope of M is defined by <TEX>$M_{\omega}$</TEX> = {x <TEX>$\in$</TEX> E(M) | Jx <TEX>$\subseteq$</TEX> M for some J <TEX>$\in$</TEX> GV (R)}. In this paper, <TEX>$\omega$</TEX>-modules over commutative rings are considered, and the theory of <TEX>$\omega$</TEX>-operations is developed for arbitrary commutative rings. As applications, we give some characterizations of <TEX>$\omega$</TEX>-Noetherian rings and Krull rings.

Module-Theoretic Characterizations of Generalized GCD Domains

We give several module-theoretic characterizations of generalized GCD domains. For example, we show that an integral domain R is a generalized GCD domain if and only if semi-divisoriality and flatness are equivalent for torsion-free R-modules if and only if every w-finite w-module is projective if and only if R is w-Prüfer (in the sense of Zafrullah). We also characterize when a pullback R of a certain type is a generalized GCD domain. As an application, we characterize when R = D + XE[X] (here, D ⊆ E is an extension of domains and X is an indeterminate) is a generalized GCD domain.

Valuation domains with a maximal immediate extension of finite rank

If R is a valuation domain of maximal ideal P with a maximal immediate extension of finite rank it is proven that there exists a finite sequence of prime ideals P = L 0 ⊃ L 1 ⊃ ⋯ ⊃ L m ⊇ 0 such that R L j / L j + 1 is almost maximal for each j, 0 ⩽ j ⩽ m − 1 and R L m is maximal if L m ≠ 0 . Then we suppose that there is an integer n ⩾ 1 such that each torsion-free R-module of finite rank is a direct sum of modules of rank at most n. By adapting Lady's methods, it is shown that n ⩽ 3 if R is almost maximal, and the converse holds if R has a maximal immediate extension of rank ⩽2.

Lie–Rinehart cohomology and integrable connections on modules of rank one

Let k be an algebraically closed field of characteristic 0, let R be a commutative k-algebra, and let M be a torsion free R-module of rank one with a connection ∇. We consider the Lie–Rinehart cohomology with values in End R ( M ) with its induced connection, and give an interpretation of this cohomology in terms of the integrable connections on M. When R is an isolated singularity of dimension d ⩾ 2 , we relate the Lie–Rinehart cohomology to the topological cohomology of the link of the singularity, and when R is a quasi-homogenous hypersurface of dimension two, we give a complete computation of the cohomology.

Direct Sum Decompositions Over Two-Dimensional Local Domains

Given a ring R and a class 𝒞 of R-modules, one can ask whether or not every element of 𝒞 decomposes uniquely as a direct sum of indecomposable elements of 𝒞. If not, one can further ask if it is possible for an element of 𝒞 to decompose both as the direct sum of s indecomposable elements of 𝒞 and as the direct sum of t indecomposable elements of 𝒞 where s ≠ t. In this article, we investigate these questions when R is a two-dimensional analytically normal domain and 𝒞 is the class of finitely generated torsion-free R-modules.

Module-Theoretic Characterizations of t-Linkative Domains

If R is an integral domain, we extend to any R-module the notion of semi-divisorial envelope, or w-envelope, defined by Wang and McCasland for torsion-free R-modules, and introduce and study the related notions of co-semi-divisoriality and w-nullity. These concepts are then used to give new module-theoretic characterizations of t-linkative domains, a class of domains widely considered in multiplicative ideal theory.

Butler Modules Over Prüfer Domains

If R is an integral domain, let 𝒞 be the class of torsion free completely decomposable R-modules of finite rank. Denote by ℛ the class of those torsion-free R-modules A such that A is a homomorphic image of some C ∈ 𝒞, and let 𝒫 be the class of R-modules K such that K is a pure submodule of some C ∈ 𝒞. Further, let Q ℛ and Q 𝒫 be the respective closures of ℛ and 𝒫 under quasi-isomorphism. In this article, it is shown that if R is a Prüfer domain, then Q ℛ = Q 𝒫, and ℛ = 𝒫 in the special case when R is h-local. Also, if R is an h-local Prüfer domain and if C ∈ 𝒞 has a linearly ordered typeset, it is established that all pure submodules and all torsion-free homomorphic images of C are themselves completely decomposable. Finally, as an application of these results, we prove that if R is an h-local Prüfer domain, then ℛ = Q ℛ = Q 𝒫 = 𝒫 if and only if R is almost maximal.

A note on the Matlis category equivalence

We provide additional information concerning the important Matlis category equivalence between the categories of h-divisible torsion and R-complete torsion-free R-modules over any domain R. We identify the h-divisible modules that correspond under this category equivalence to the Enochs cotorsion torsion-free modules as the weak-injective modules. As a consequence, we can characterize weak-injective modules in terms of their flat covers. We also determine the range of the injective dimensions of pure-injective modules.

Gruson-Jensen Duality for Idempotent Rings

ABSTRACT It is known that if R is a ring with identity, and S and A op are the functor rings associated to the categories Mod(R) and Mod(R op ), respectively, then there is a duality between the categories of finitely presented objects of Mod(S op ) and Mod(A). We prove here this result in a more general case, namely when R is an idempotent ring, not necessarily having an identity, and when the categories Mod(R) of torsionfree and unitary right R-modules and Mod(R op ) of torsionfree and unitary left R-modules are locally finitely presented.

Minimal modules over valuation domains

Let R be a valuation domain. We say that a torsion-free R-module is minimal if it is isomorphic to all its submodules of finite index. Here, the usual concept of finite index for groups is replaced by the more appropriate (for module theory) definition: a submodule H of the module G is said to be of finite index in G if the quotient G / H is a finitely presented torsion module. We investigate minimality in various settings and show inter alia that over a maximal valuation domain, all torsion-free modules are minimal. Constructions of non-minimal modules are given by utilizing realization theorems of May and the authors. We also show that direct sums of minimal modules may fail to be minimal.

Direct sum cancellation for modules over one-dimensional rings

Let R be a one-dimensional Noetherian domain with finite normalization R. In the eighties the second-named author and S. Wiegand developed a mechanism for studying the cancellation problem for finitely generated torsion-free R-modules. The key idea, described in [Wie84] and [WW87], is to represent a given torsion-free module M as a pullback: M −→ RM ↓ ↓ M/fM −→ RM/fM ∗This research was initiated while the author was a visitor at the University of Nebraska, supported by the Fonds zur Forderung der wissenschaftlichen Forschung. †Partially supported by grants from the National Science Foundation, the National Security Agency and the Mathematical Sciences Research Institute, where much of this research was done.

Strongly torsion free, copure flat and Matlis reflexive modules

In this paper we show that if R is a ring of Krull dimension d and M is a strongly copure flat (respectively, strongly copure injective) module, then E⊗ R M (respectively, Hom R ( E, M)) is strongly cotorsion (respectively, strongly torsion free) for any injective module E. We obtain a new characterization for copure flat modules. We prove that M is copure flat if and only if Ext R 1( M, F)=0 for all flat cotorsion modules F. Moreover, if ( R, m ) is a Cohen–Macaulay local ring and M is strongly copure flat, then Hom R ( M, F) is strongly torsion free. Let ( R, m ) be a Cohen–Macaulay local ring of Krull dimension d and M be a Matlis reflexive strongly torsion free R-module. We show that M ̂ is a maximal Cohen–Macaulay R ̂ -module. Also, if R is as above and M a Matlis reflexive strongly copure injective R-module, then Hom R(E(R/ m),M) is either zero or a maximal Cohen–Macaulay R ̂ -module.

Finite representation type and direct-sum cancellation

Consider the notion of finite representation type (FRT for short): An integral domain R has FRT if there are only finitely many isomorphism classes of indecomposable finitely generated torsion-free R-modules. Now specialize: Let R be of the form D+c O where D is a principal ideal domain whose residue fields are finite, c∈ D is a nonzero nonunit, and O is the ring of integers of some finite separable field extension of the quotient field of D. If the D-rank of R is at least four then R does not have FRT. In this case we show that cancellation of finitely generated torsion-free R-modules is valid if and only if every unit of O/c O is liftable to a unit of O . We also give a complete analysis of cancellation for some rings of the form D+c O having FRT. We include some examples which illustrate the difficult cubic case.

Splitting off free summands of torsion-free modules over complete DVRs

If R is a complete discrete valuation ring and M is a reduced, torsion-free R-module of rank \kappa, where \aleph_0 \leq \kappa &lt; 2^ (\aleph_0), we show that M \prop\oplus_(\aleph_0) R \oplus C for some R-module C. As a consequence, it must be the case that M \prop M \oplus (\oplus{_\alpha}R), where \alpha \leq \aleph_0, and {\rm (End)_R}M/\rm (Fin)M has rank at least 2^ (\aleph_0), where Fin M denotes the set of endomorphisms of M with finite rank image.