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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.

We construct infinite families of pairwise non-isomorphic indecomposable totally reflexive modules of high multiplicity. Under suitable conditions on the totally reflexive modules M and N, we find infinitely many non-isomorphic indecomposable modules arising as extensions of M by N. The construction uses the bimodule structure of \({Ext^{1}_{R}}((M,N)\) over the endomorphism rings of N and M. Our results compare with a recent theorem of Celikbas, Gheibi and Takahashi, and broaden the scope of that theorem.

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.

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 prove that neat and coneat submodules of a module coincide when $R$ is a commutative ring such that every maximal ideal is principal, extending a recent result by Fuchs. We characterise absolutely neat (coneat) modules and study their closure properties. We show that a module is absolutely neat if and only if it is injective with respect to the Dickson torsion theory.

This paper introduces the notion of a functorial torsion class (FTC): in a concrete category $\mathfrak{C}$ which has image factorization, one considers monocoreflective subcategories which are closed under formation of subobjects. Here the interest is in FTCs in the category of abelian lattice-ordered groups with designated strong order unit. The FTCs $\mathfrak{T}$ consisting of archimedean latticeordered groups are characterized: for each subgroup A of the rationals with the identity 1, either $\mathfrak{T} = \mathfrak{S}\left( A \right)$, the class of all lattice-ordered groups of functions on a set X which have finite range in A, or $$\mathfrak{T} = \mathbb{T}\left( A \right)$$, the class of all subgroups of A with 1. As for FTCs possessing non-archimedean groups, it is shown that if $\mathfrak{T}$ is an FTC containing a subgroup A of the reals with 1, of rank two or greater, then $\mathfrak{T}$ contains all ℓ-groups of the form $A\vec \times G$, for all abelian lattice-ordered groups G. Finally, the least FTC that contains a non-archimedean group is the class of all $\mathbb{Z}\vec \times G$, for all abelian lattice-ordered groups G.

In this paper, we study w-nullity and (co-)semi-divisoriality of Hom-modules and the semi-divisorial envelope of HomR(M;N) under suitable conditions on R;M; and N. We also investigate an injective cogenerator of a quotient category.

A module M over a ring R is said to satisfy ( P ) if every gen-erating set of M contains an independent generating set. The followingresults are proved;(1) Let τ = (T τ , F τ ) be a hereditary torsion theory such that T τ 6=Mod-R . Then every τ -torsionfree R -module satisfies ( P ) if and only if S = R/τ ( R ) is a division ring.(2) Let K be a hereditary pre-torsion class of modules. Then everymodule in K satisfies ( P ) if and only if either K = { 0 } or S = R/ Soc K ( R )is a division ring, where Soc K ( R ) = ∩{I ≤ R R : R/I ∈ K} . For a right R -module M , a subset X of M is said to be a generating set of M if M = Σ x∈X xR ; and a minimal generating set of M is any generating set Y of M such that no proper subset of Y can generate M . A generating set X of M is called an independent generating set if Σ x∈X xR = ⊕ x∈X xR . Clearly, everyindependent generating set of M is a minimal generating set, but the converseis not true in general. For example, the set { 2 , 3 } is a minimal generating setof Z

A Grothendieck category C is said to be locally finitely generated if the subobject lattice of every object in C is compactly generated, or equivalently, if C possesses a family of finitely generated generators. Every nonzero locally finitely generated Grothendieck category possesses simple objects. We shall call a Grothendieck category C indecomposable if C is not equivalent to a product of nonzero Grothendieck categories C1×C2. In this paper an example of an indecomposable nonlocally finitely generated Grothendieck category possessing simple objects is constructed, answering in the negative a sharper form of a question posed by Albu, Iosif, and Teply in [T. Albu, M. Iosif, M.L. Teply, Dual Krull dimension and quotient finite dimensionality, J. Algebra 284 (2005) 52–79].

application/pdf, We show that if A is a representation-finite selfinjective Artin algebra, then every P• ∈ Kb(PA) with HomK(Mod-A)(P•,P•[i]) = 0 for i ≠ 0 and add(P•) = add(νP•) is a direct summand of a tilting complex, and that if A, B are derived equivalent representation-finite selfinjective Artin algebras, then there exists a sequence of selfinjective Artin algebras A = B0, B1,…, Bm = B such that, for any 0 ≤ i < m, Bi+1 is the endomorphism algebra of a tilting complex for Bi of length ≤ 1.

In this note we give a new and elementary proof of a result of Nastasescu and Torrecillas (J. Algebra, 281:144–149, 2004) stating that a coalgebra C is finite dimensional if and only if the rational part of any right module M over the dual algebra \(C^*\) is a direct summand in M (the splitting problem for coalgebras).

Letτbe a hereditary torsion theory onModRand suppose thatQτ:ModR→ModRis the localization functor. It is shown that for allR-modulesM, every higher derivation defined onMcan be extended uniquely to a higher derivation defined onQτ(M)if and only ifτis a higher differential torsion theory. It is also shown that ifτis a TTF theory andCτ:M→Mis the colocalization functor, then a higher derivation defined onMcan be lifted uniquely to a higher derivation defined onCτ(M).

Let S be a ring extension of R, and let τ R =( T R , F R ) and τ S =( T S , F S ) be torsion theories for right R-modules and right S-modules respectively. Using functors HomR(−,S) and −⊗RS, we get the connection between τR and τS, introduce torsion theory extensions related to ring extensions and establish some relations between finite normalizing extensions and torsion theory extensions. Furthermore, we consider some applications of torsion theory extensions and generalize some results of excellent extensions (almost excellent extensions) to the setting of torsion theories.

Using strong equivalences for coalgebras we define the strong Brauer group of a cocommutative coalgebra C, which is a subgroup of the Brauer group of C. In general there is not a good relation between the Brauer group of a coalgebra and the Brauer group of the dual algebra C-*, the former is not even a torsion group. We find that this subgroups embeds in the Brauer group of C-*. A key tool in this result is the use of techniques from torsion theory. Some cases where both subgroups coincide are shown, for example, C being coreflexive. (C) 2002 Elsevier Science B.V. All rights reserved.

Let \(\sigma = ({\mathcal{T}},{\mathcal{F}})\) be a hereditary torsion theory for the category \({\mathcal{R}}\)-mod of unital left \({\mathcal{R}}\)-modules over an associative ring \({\mathcal{R}}\) with an identity element. The purpose of this note is to prove that if the associated Gabriel filter \({\mathcal{L}}\) consists of finitely presented left ideals, then every module has a \(\sigma \)-injective cover and if \({\mathcal{L}}\) contains a cofinal subset of finitely presented left ideals, then every module has a \(\sigma \)-torsionfree \(\sigma \)-injective cover. The methods used working with pure submodules contained in ``large" submodules also allow to unify the proofs of some previously known 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).

Necessary and sufficient conditions are provided for the nilpotence of the intersection of all τ-closed prime ideals of a ring R with τ-Krull dimension, where τ denotes a hereditary torsion theory on the category Mod-R of right R-modules. It is shown that these conditions hold, in particular, when τ is ideal invariant. Furthermore, conditions are provided for Ass(M) to be non-empty for every τ-torsionfree right R-module M ≠ 0.

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.

We extend the existing concepts of secondary representation of a module, coregular sequence and attached prime ideals to the more general setting of any hereditary torsion theory. We prove that any $\tau$-artinian module is $\tau$-representable and that such a representation has some sort of unicity in terms of the set of $\tau$-attached prime ideals associated to it. Then we use $\tau$-coregular sequences to find a nice way to compute the relative width of a module. Finally we give some connections with the relative local homology.

We generalize the well-known fact that for a pair of Morita equivalent ringsR andS their maximal rings of quotients are again Morita equivalent: If τn (M) denotes the torsion theory cogenerated by the direct sum of the firstn+1 injective modules forming part of the minimal injective resolution ofM then ατn (R)=τn (S) where α is the category equivalenceR-Mod→S-Mod. Consequently the localized ringsRτn(R) andSτn(S) are Morita equivalent.

Let [Formula: see text] be an arbitrary ring with identity and [Formula: see text] a right [Formula: see text]-module with the ring [Formula: see text] End[Formula: see text] of endomorphisms of [Formula: see text]. The notion of an [Formula: see text]-inverse split module [Formula: see text], where [Formula: see text] is a fully invariant submodule of [Formula: see text], is defined and studied by the present authors. This concept produces Rickart submodules of modules in the sense of Lee, Rizvi and Roman. In this paper, we consider the submodule [Formula: see text] of [Formula: see text] as [Formula: see text] and [Formula: see text], and investigate some properties of [Formula: see text]-inverse split modules and [Formula: see text]-inverse split modules [Formula: see text]. Results are applied to characterize rings [Formula: see text] for which every free (projective) right [Formula: see text]-module [Formula: see text] is [Formula: see text]-inverse split for the preradicals such as [Formula: see text] and [Formula: see text].

Let $\Lambda$ be an artinian ring. Generalizing the Loewy length, we propose the layer length associated with a torsion theory, which is a new measure for finitely generated $\Lambda$-modules., Comment: 12 pages

We define coherence relative to an arbitrary torsion theory and characterize it in terms of preenvelopes of modules. In particular, some known results are obtained as corollaries.

We extend some recent results on the differentiability of torsion theories. In particular, we generalize the concept of (α, β)-derivation to (α, β)-higher derivation and demonstrate that a filter of a hereditary torsion theory that is invariant for α and β is (α, β)-higher derivation invariant. As a consequence, any higher derivation can be extended from a module to its module of quotients. Then, we show that any higher derivation extended to a module of quotients extends also to a module of quotients with respect to a larger torsion theory in such a way that these extensions agree. We also demonstrate these results hold for symmetric filters as well. We finish the paper with answers to two questions posed in [L. Vas, Extending higher derivations to rings and modules of quotients, International Journal of Algebra, 2 (15) (2008), 711–731]. In particular, we present an example of a non-hereditary torsion theory that is not differential.

We characterize the hereditary torsion pairs of finite type in the functor category of a ring R that are associated to tilting torsion pairs in the category of R-modules. Moreover, we determine a condition under which they give rise to TTF triples.

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.

We define and study the symmetric version of differential torsion theories. We prove that the symmetric versions of some of the existing results on derivations on right modules of quotients hold for derivations on symmetric modules of quotients. In particular, we prove that the symmetric Lambek, Goldie, and perfect torsion theories are differential. We also study conditions under which a derivation on a right or symmetric module of quotients extends to a right or symmetric module of quotients with respect to a larger torsion theory. Using these results, we study extensions of ring derivations to maximal, total, and perfect right and symmetric rings of quotients.

The dimension of any module over an algebra of affiliated operators ${\mathcal U}$ of a finite von Neumann algebra ${\mathcal A}$ is defined using a trace on ${\mathcal A}.$ All zero-dimensional ${\mathcal U}$-modules constitute the torsion class of torsion theory $(\mathrm{{\bf T}},\mathrm{{\bf P}})$. We show that every finitely generated ${\mathcal U}$-module splits as the direct sum of torsion and torsion-free part. Moreover, we prove that the theory $(\mathrm{{\bf T}},\mathrm{{\bf P}})$ coincides with the theory of bounded and unbounded modules and also with the Lambek and Goldie torsion theories. Lastly, we use the introduced torsion theories to give the necessary and sufficient conditions for ${\mathcal U}$ to be semisimple., Comment: To appear in Rocky Mountain Journal of Mathematics

In this paper, we introduce and investigate the latticial counterparts of the conditions (Ci), i = 1, 2, 3, 11, 12, for modules. In particular, we study the lattices satisfying the condition (C1), we call CC lattices (for Closed are Complements), i.e. the lattices such that any closed element is a complement, that are the latticial counterparts of CS modules (for Closed are Summands). Applications of these results are given to Grothendieck categories and module categories equipped with a torsion theory.

In this survey article, we indicate how some of the recent ideas and techniques introduced in the study of infinite rank Butler groups can be successfully used in the investigation of the homological dimensions of torsion-free modules over integral domains and, in particular, over valuation domains.

E. Enochs began the study of injective covers in [3],characterizing when any lefti?-module has an injectivecover. This happens ifand onlyifR is aleft noetherian ring. Torsion injective covers and torsionfreeinjective covers were introduced by Ahsan and Enochs in [2]and [1]respectively,in the context of the Goldie torsion theory. B. Torrecillasin [13] defined r-torsionfree r-injective covers and r-injectivecovers for x a hereditary torsion theory. These covers have been studied in [6] and [7]. In thispaper, r-torsion r-injectivecovers and envelopes are studied for x any torsion theory. Then we construct relativehomological algebra by means of complexes with thiskind of covers and envelopes. In Section 3, we find necessary and sufficientconditions for the existence of r-torsion r-injective covers and envelopes forany module. In the aim of thedescomposition theorem of abelian groups in divisibleand reduced part, we give a torsion theoretical version in terms of r-torsion r-injectivemodules as divisiblesones, Proposition 2, relating such descomposition with certain condition on the existence of r-torsion r-injectivecovers. The existence of r-torsion r-injectiveenvelopes is given in Theorem 2. In Section 4, we study the balance (see [5]) of the functor Hom(―,―) relativeto theclass of r-torsionr-injectivemodules. When thebalance is given, itispossible tointroduce leftderived functor of Hom(―, ―)by using resolvents and resolutions of r-torsion r-injectivemodules. Left and right relativeglobal dimension of the ring R are defined and analysed.

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.

IfM is a module torsionfree and noetherian with respect to a torsion theory, ifS is the endomorphism ring ofM, and ifL(S) is the ideal ofS consisting of all endomorphisms with large kernels, thenL(S) is nilpotent and a bound on the index of nilpotency ofL(S) is given.

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].

A torsion theory partitions the spectrum of the base ring into two sets. Over a Noetherian ring, every suitable partition of the spectrum gives rise to one and only one torsion theory. It is possible to know whether a module is torsion or torsion-free by looking at its associated primes. The example of the polynomial torsion theory is developed. Notations and terminology. All rings are commutative with 1. We denote by AssA(M) the set of weakly associated primes of an A-module M. A prime p is in ASSA(M) if p is minimal among the primes containing the annihilator annA(x) of a nonzero element x of M. We denote by AssA(M) the set of strongly associated primes of an Amodule M. A prime p is in AssA(M) if it is the annihilator annA(x) of a nonzero element x of M. We denote by SuppA(M) the support of an A-module M. A prime p is in SUPPA(M) if the localization MJ is not null. We send the reader to Lambek [5] for the definition and immediate properties of a torsion theory. We denote by T(M) the torsion submodule of a module M. 1. Partition of the spectrum. Throughout this section A is a ring with a torsion theory. PROPOSITION 1. 1. Let M be be an A-module. (1) If A/p is not a torsion module, for every p in ASSA(M), then M is a torsion-free module. (2) If M is a torsion-free module, then A/p is torsion-free for every p in AssA(M). PROOF. (1) Suppose that M is not a torsion-free module. The torsion submodule T(M) must contain a nonzero element x and AIannA(x)-Ax, Presented to the Society, May 1, 1972; received by the editors May 5, 1972. AMS (MOS) subject classifications (1970). Primary 13C10.

Given two ringsRandS, we study the category equivalences T ⇄ Y , where T is a torsion class ofR-modules and Y is a torsion-free class ofS-modules. These equivalences correspond to quasi-tilting triples (R, V, S), whereRVSis a bimodule which has, “locally,” a tilting behavior. Comparing this setting with tilting bimodules and, more generally, with the torsion theory counter equivalences introduced by Colby and Fuller, we prove a local version of the Tilting Theorem for quasi-tilting triples. A whole section is devoted to examples in case of algebras over a field.

First, we show that a compact object $C$ in a triangulated category, which satisfies suitable conditions, induces a $t$-structure. Second, in an abelian category we show that a complex $P^{\centerdot}$ of small projective objects of term length two, which satisfies suitable conditions, induces a torsion theory. In the case of module categories, using a torsion theory, we give equivalent conditions for $P^{\centerdot}$ to be a tilting complex. Finally, in the case of artin algebras, we give one to one correspondence between tilting complexes of term length two and torsion theories with certain conditions., 17 pages, AMSLaTeX

A modular lattice L with 0 and 1 is called quotient finite dimensional (QFD) if [x,1] has no infinite independent set for any x∈L. We characterize upper continuous modular lattices L that have dual Krull dimension k0(L)⩽α, by relating that with the property of L being QFD and with other conditions involving subdirectly irreducible lattices and/or meet irreducible elements. In particular, we answer in the positive, in the more general latticial setting, some open questions on QFD modules raised by Albu and Rizvi [Comm. Algebra 29 (2001) 1909–1928]. Applications of these results are given to Grothendieck categories and module categories equipped with a torsion theory.

The purpose of this note is to show that the only idempotent kernel functor σ on Mod-R, whereR is a semiprime ring andR R has finite σ-length, is the idempotent kernel functorZ corresponding to the Goldie torsion theory.

In this note, we show that if a topologyF¯over a ringAsatisfies a certain finiteness condition, then the Gabriel topologyG¯generated byF¯can be explicitly constructed and it also satisfies the same finiteness condition.

Composition series for modules with respect to a Gabriel topology (or equivalently, with respect to a hereditary torsion theory) were introduced by Goldman [1] in 1975, but only for torsion-free modules, and have been further studied, among others, by Beachy, Golan,etc.

We consider a generalization of lifting modules relative to a class [Formula: see text] of modules and a proper class 𝔼 of short exact sequences of modules. These modules will be called 𝔼-[Formula: see text]-lifting. We establish characterizations of modules with the property that every direct sum of copies of them is 𝔼-[Formula: see text]-lifting.