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Let R be a commutative ring with identity and τ be a hereditary torsion theory on R-Mod. In this article, we introduce and study the concept of τ-comultiplication module. We present several propert...

We give a classification of torsion classes (or nullity classes) in an abelian category by forming a spectrum of equivalence classes of premonoform objects. This is parallel to Kanda’s clas...

In this article, we study the hereditary torsion theory defined by the set of associated primes of principle ideals of an integral domain, which is called the g-torsion theory. We first discuss some general properties of g-torsion theories, and after that give some applications of them. For example, we generalize a characterization of reflexive modules over quasi-normal domains to a class of non-Noetherian domains. Among other things, a characterization of coherent domains of weak Gorenstein global dimension at most two is also given in terms of Gorenstein projectivity (or Gorenstein flatness) of injective modules relative to the g-torsion theory.

Using the concept of prime submodule for M ∈ R-Mod, P ∈ Spec(M), and N ∈ σ[M], we define when N is P-Mtame \ (Mtame) module. This concept generalizes the concept\ of P-tame (tame) modules. For M ∈ R-Mod and τ ∈M-tors, we use the concept of τ M -Gabriel dimension and we study the relationship between Mtame modules and τ M -Gabriel dimension. We find equivalent conditions for a module M progenerator in σ[M] with τ M -Gabriel dimension to have τ M -Gabriel correspondence in terms of the P-Mtame modules. This result extends the results by Albu et al. and Kim and Krause.

In the present article, we analyze a d36-effect piezoelectric torsion transducer following the Saint-Venant torsion theory taking the electrical field into account. A representation of the stress function, the electric potential, and the warping function are derived and solved with finite differences. Then, the one-dimensional governing equations at the structural beam level, including the constitutive relations as well as the balance equations for the dynamics of the transducer, are presented. The axial moment and the total charge are computed as functions of the rate of twist and the applied potential difference. As an example, a cantilevered transducer is studied.

Using the nice properties of the w-divisible weight and the w-divisible groups, we prove a factorization theorem for compact abelian groups K; namely, K = K tor × K d , where K tor is a bounded torsion compact abelian group and K d is a w-divisible compact abelian group. By Pontryagin duality this result is equivalent to the same factorization for discrete abelian groups proved in [9].

It is well-known (see [13]) that a hereditary torsion theory τ for the category R-mod is noetherian if and only if the class of all τ-torsionfree τ-injective modules is closed under arbitrary direct sums. So, it is natural to investigate the hereditary torsion theories having the property that the class of all τ-torsionfree injective modules is closed under arbitrary direct sums, which are called ℱ-noetherian. These torsion theories have been studied by Teply in [16]. In the second part of this note we shall study the weakly exact hereditary torsion theories, which generalize the exact one's.

We consider the concept of prime submodule defined by Raggi et al. [7]. We find equivalent conditions for a module M progenerator in σ[M], with τ M -Gabriel dimension, to have a one-to-one correspondence between the set of isomorphism classes of indecomposable τ-torsion free injective modules in σ[M] and the set of τ-pure submodules prime in M, where τ is a hereditary torsion theory in σ[M]. Also we give a relation between the concept of prime M-ideal given by Beachy and the concept of prime submodule in M. We obtain that if M is progenerator in σ[M], then these concepts are equivalent.

The aim of this article is to present a latticial version of the renown module theoretical Osofsky–Smith Theorem.

Let R be a right perfect ring, and let (ℱ, 𝒞) be a cotorsion theory in the category of right R-modules ℳ R . In this article, it is shown that every right R-module has a superfluous ℱ-cover if and only if there exists a torsion theory (𝒜, ℬ) such that (ℱ, 𝒞) is cogenerated by ℬ. It is also proved that if (𝒜, ℬ) is a cosplitting torsion theory, then (⊥ℬ, (⊥ℬ)⊥) is a hereditary and complete cotorsion theory, and if (𝒜, ℬ) is a centrally splitting torsion theory, then (⊥ℬ, (⊥ℬ)⊥) is a hereditary and perfect cotorsion theory.

Let R be a ring. A right R-module U is called Tor-tilting if , where U + = Homℤ(U, ℚ/ℤ), Cogen(U +) is the class of left R-modules cogenerated by U + and consists of modules R M such that . Some examples and characterizations of Tor-tilting modules are given. Among others, it is shown that U R is Tor-tilting if and only if U + is cotilting. Moreover, both tilting modules and completely faithful flat modules are proved to be Tor-tilting. The torsion theory induced by a Tor-tilting module is also investigated.

Many observations about coalgebras were inspired by comparable situations for algebras. Despite the prominent role of prime algebras, the theory of a corresponding notion for coalgebras was not well understood so far. Coalgebras C over fields may be called coprime provided the dual algebra C* is prime. This definition, however, is not intrinsic—it strongly depends on the base ring being a field. The purpose of the article is to provide a better understanding of related notions for coalgebras over commutative rings by employing traditional methods from (co)module theory, in particular (pre)torsion theory. Dualizing classical primeness condition, coprimeness can be defined for modules and algebras. These notions are developed for modules and then applied to comodules. We consider prime and coprime, fully prime and fully coprime, strongly prime and strongly coprime modules and comodules. In particular, we obtain various characterisations of prime and coprime coalgebras over rings and fields.

Let M be a module and K a submodule of a module N in σ[M]. We call K a δ–M-small submodule of N if whenever N = K + L, N/L is M-singular for any submodule L of N, we have N = L. Also we call N a δ–M-small module if N is a δ–M-small submodule of its M-injective hull. In this article, we consider , the reject of  ℳ in N, where  ℳ is the class of all δ–M-small modules. We investigate the properties of and consider the torsion theory τδ V in σ[M] cogenerated by  ℳ. We compare the τδ V and the torsion theory τ V cogenerated by M-small modules and finally we give a characterization of GCO-modules.

Let M R be a right R-module over a ring R with S = End(M R ). We study the coherence of the left S-module S M relative to a hereditary torsion theory for the category of right R-modules. Various results are developed, many extending known results.

The article presents a theoretical survey of open profile reinforced concrete bent beams behaviour in mixed torsion. A model, based on opened thin profile beams mixed torsion theory, permitted to take account of the concrete stretched for the determination of the torsion's characteristics of the profile according to the state of longitudinal strain due to bending, and showed that the level of the flexural cracking is an important parameter, its growth influences meaningfully on these features. The results descended of the model are practically in conformity with the experimental results.

Let M and N be R-modules. We define where S denotes the class of all M-small modules. We call N an M-cosingular (non-M-cosingular) module if Z M (N) = 0 ( Z M (N) = N). We study the properties of M-cosingular and non-M-cosingular modules in σ[M] We consider the torsion theory cogenerated by M-small modules and show that it is cohereditary when every injective module in σ[M] is amply supplemented. We also give instances where this torsion theory is cohereditary or splits. Finally we characterise lifting modules N ∊ [M] in terms of Z 2 M (N).

Conditions under which a *-module AU extends to a *-module RR⊗AU over a ring extension R of A are presented, along with con¬nections to quasi-progenerators, tilting modules, and torsion theory counter equivalences.

This article introduces the concept of μ-complemented modules as follows: given a hereditary torsion theory in Mod-R with associated torsion functor μ we say that a module M is μ-complemented when every sub module is μ- dense in a summand of M. We present here some fundamental properties of this class of modules and study the relationship between them and extending modules. We apply these results, in particular, when μ is the torsion functor associated to the Goldie torsion theory and when μ is jansian. The question of which direct sums of μ-complemented R-modules are μ-complemented is treated here. We also obtain decomposition theorems for μ-complemented modules when either the ring or the module satisfies certain chain conditions.

Let M be an R-module and α[M] the category of all R-modules that are subgenerated by M. For any hereditary torsion theory T in α[M], the class of quotient modules defines a Grothendieck category denoted by ∊T[M]. T is called spectral if ∊T[M] is a spectral category (every short exact sequence in∊T[M] splits). The paper is devoted to the investigation of such torsion theories. Various characterizations are obtained and properties of the endomorphism rings of the corresponding quotient modules are studied. The final sectlon is concerned with the role of simple and semisimple modules with respect to spectral torsion theories. Our presentation generalizes and extends related results for spectral torsion theories in the category of all R-modules R-Mod.

In a previous paper we introduced μ-complemented modules for any given torsion functor μ of a hereditary torsion theory in Mod-R and presented their basic properties. Our current research enables us to characterize these modules when μ is a proper torsion theory other Than Goldie's and the ring is a one-dimensional Noetherian domain. The case in which μ has property (T) is also treated. For an arbitrary μ, rings for which every module is μ-complemented are considered.

(1995). Tilting and torsion theory counter equivalences. Communications in Algebra: Vol. 23, No. 13, pp. 4833-4849.

We establish relations between two extensions of Dickson's torsion theory, one introduced by Barr and the other by Cassidy, Hubert and Kelly. These two notions are studied in a general category and we show that categories with zero object are, in some sense, the ones where the second notion is relevant. In these categories, under suitable completeness conditions, we characterize torsion theories in terms of reflections and connections. A characterization of the torsion-free subcategories which generalizes the one of Dickson is also presented.

For a torsion radical, σ, σ-flatness (as defined in [8]) is studied. Many of the properties of flat modules are shown to extend and σ-flat modules are characterized under certain restrictions on the associated filter, Lσ. We also define semi-σ-flat modules in a natural way and find conditions equivalent to every R-module being semi-σ-flat.

A notion of closure operator for modules is used to characterize factorization structures in categories of modules. Moreover compactness, injectivity and absolute closedness are studied with respect to such closure operators. A criterion for compactness of modules is obtained in terms of injectivity or absolute closedness of the quotients extending recent results of Temple Fay.

Let R be an associative ring with 1 and let T be a hereditary torsion theory in the category of left R-modules. In defining the localizatio n of R respect to x, the concept of T-injective module arises (see [5] , [11]) . We can consider the family E T of all short exact sequences of left R-modules respect to which every T-injective left R-module is injective . E T proper class in the sense defined in [ 9] . In this paper we characterize proper classe s which are of the form E T for some hereditary torsion theory x. On the other hand, we give some conditions on x, which imply that E T has enough projectives , and we show an example where E T does not have enough projectives.

This paper continues the study of codivisible modules, whose definition is a “dualization” of Lambek's concept [4] of a divisible module relative to a torsion theory. The main purpose of this work is to give a solution to the following problem posed by Bland [2]: “It would be interesting to know under what conditions the universal existence of codivisible covers implies that of projective covers.” The if-and-only-if nature of the solution, which is given in Theorems 2 and 4, shows that our sufficient conditions are “best possible” conditions. The method of the solution introduces the concept of a pseudo-hereditary torsion theory, which may be of interest in its own right; in particular, every hereditary torsion theory and every faithful torsion theory is pseudo-hereditary. The main results for a pseudo-hereditary torsion theory (T, F) relate the conditions, R/T(R) is semiperfect and R/T(R) is left perfect, to the existence of codivisible covers (see Theorems 1 and 3 and Corollaries 1 and 2).

In the first section we generalize the concept of the socle of a module by replacing simples with τ-simple modules for a hereditary torsion theory τ. The second section is concerned with the τ-Loewy series, and finally these general results are applied in the section 3 to the notions of τ-semiartinian rings and modules.

The concept of cotorsion was first introduced in the category of Abelian groups (Fuchs [l] ). Matlis [5], studied the cotorsion modules over integral domains. Henderson and Orzech [4], Fuchs [2], and Mines [6], replaced the classical notion of torsion by a torsion theory (T,F) on R-mod, where R is not necessarily commutative ring. In this paper we find conditions on the torsion theory in order to get a T-cotorsion hull for every module. This generalizes the result of Fuchs [2].

Simmons [1986] has defined a new filtration for torsion theories over module categories. In this note we show that this filtration coincides with the Gabriel filtration for a large class of rings.

In this work we obtain a generalization and a justification of Saint Venant's torsion theory in beams, by applying the asymptotic expansion method to the Hellinger-Reissner variational formulation of the three dimensional linearized elasticity model