Hereditary Torsion Theory Research Articles

On torsion theories and open classes of linear modular lattices

Given a ring R, a bijection exists between torsion theories and idempotent radicals. Further, the class of hereditary torsion theories contains the skeleton of the open classes. In this work, we extend the notions and main results in R-Mod about torsion classes, torsion-free classes, and open classes to the category of linear lattices, whose objects are complete modular lattices and whose morphisms are linear morphisms.

Gradual and Fuzzy Modules: Functor Categories

The categorical treatment of fuzzy modules presents some problems, due to the well known fact that the category of fuzzy modules is not abelian, and even not normal. Our aim is to give a representation of the category of fuzzy modules inside a generalized category of modules, in fact, a functor category, Mod−P, which is a Grothendieck category. To do that, first we consider the preadditive category P, defined by the interval P=(0,1], to build a torsionfree class J in Mod−P, and a hereditary torsion theory in Mod−P, to finally identify equivalence classes of fuzzy submodules of a module M with F-pair, which are pair (G,F), of decreasing gradual submodules of M, where G belongs to J, satisfying G=Fd, and ∪αF(α) is a disjoint union of F(1) and F(α)\G(α), where α is running in (0,1].

Hereditary Torsion Theories for Graphs

Using congruences, a Hoehnke radical can be defined for graphs in the same way as for universal algebras. This leads in a natural way to the connectednesses and disconnectednesses (= radical and semisimple classes) of graphs. It thus makes sense to talk about ideal-hereditary Hoehnke radicals for graphs (= hereditary torsion theories). All such radicals for the category of undirected graphs which allow loops are explicitly determined. Moreover, in contrast to what is the case for the well-known algebraic categories, it is shown here that such radicals for graphs need not be Kurosh–Amitsur radicals.

On the nilpotence of the prime radical in module categories

For M∈R-Mod and τ a hereditary torsion theory on the category σ[M] we use the concept of prime and semiprime module defined by Raggi et al. to introduce the concept of τ-pure prime radical Nτ(M)=Nτ as the intersection of all τ-pure prime submodules of M. We give necessary and sufficient conditions for the τ-nilpotence of Nτ(M). We prove that if M is a finitely generated R-module, progenerator in σ[M] and χ≠τ is FIS-invariant torsion theory such that M has τ-Krull dimension, then Nτ is τ-nilpotent.

An extension of S-artinian rings and modules to a hereditary torsion theory setting

For any commutative ring A, we introduce a generalization of S-artinian rings using a hereditary torsion theory σ instead of a multiplicative closed subset It is proved that if A is a totally σ-artinian ring, then σ must be of finite type, and A is totally σ-noetherian.

A characterization of Prüfer $v$-multiplication domains in terms of linear equations

In this paper, we characterize Prüfer v-multiplication domains as integral domains which have the property that the existence of a generalized solution of any system of linear equations is equivalent to a weak equality of the determinantal ideals of the coefficient matrix and the augmented matrix of the system. In fact, we obtain a more general result for commutative rings of weak global τ-dimension (in the sense of Bueso, Van Ostaeyen and Verschoren) at most one, where τ is a half-centered hereditary torsion theory.

N-Coherence Relative to a Hereditary Torsion Theory

Let R be a ring, τ=T,ℱ a hereditary torsion theory of mod-R, and n a positive integer. Then, R is called right τ-n-coherent if every n-presented right R-module is τ,n+1-presented. We present some characterizations of right τ-n-coherent rings, as corollaries, and some characterizations of right n-coherent rings and right τ-coherent rings are obtained.

Torsion Pairs in Categories of Modules over a Preadditive Category

It is a result of Gabriel that hereditary torsion pairs in categories of modules are in bijection with certain filters of ideals of the base ring, called Gabriel filters or Gabriel topologies. A result of Jans shows that this bijection restricts to a correspondence between (Gabriel filters that are uniquely determined by) idempotent ideals and TTF triples. Over the years, these classical results have been extended in several different directions. In this paper, we present a detailed and self-contained exposition of an extension of the above bijective correspondences to additive functor categories over small preadditive categories. In this context, we also show how to deduce parametrizations of hereditary torsion theories of finite type, Abelian recollements by functor categories, and centrally splitting TTFs.

A new semistar operation on a commutative ring and its applications

In this article, a new semistar operation, called the q-operation, on a commutative ring R is introduced in terms of the ring of finite fractions. It is defined as the map by there exists some finitely generated semiregular ideal J of R such that for any where denotes the set of nonzero R-submodules of The main superiority of this semistar operation is that it can also act on R-modules. We can also get a new hereditary torsion theory τq induced by a (Gabriel) topology is an ideal of R with Based on the existing literature of τq-Noetherian rings by Golan and Bland et al., in terms of the q-operation, we can study them in more detailed and deep module-theoretic point of view, such as τq-analog of the Hilbert basis theorem, Krull’s principal ideal theorem, Cartan-Eilenberg-Bass theorem, and Krull intersection theorem.

A new version of a theorem of Kaplansky

A well-known theorem of Kaplansky states that any projective module is a direct sum of countably generated modules. In this paper, we prove the w-version of this theorem, where w is a hereditary torsion theory for modules over a commutative ring.Communicated by Silvana Bazzoni

Big lattices of module classes induced by preradicals

We introduce new big lattices of classes of R-modules induced by suitable preradicals, the big lattice of $$\sigma $$-hereditary classes, the big lattice of$$\sigma $$-cohereditary classes, and the big lattice of $$\sigma $$-open classes. We prove that their Skeletons are boolean lattices which generalize and extend the lattice of natural classes, the lattice of conatural classes and the Skeleton of the lattice of hereditary torsion theories, respectively. We also introduce the lattice of$$\sigma $$-hereditary torsion theories, for a exact preradical $$\sigma $$. In case that $$\sigma $$ be the identity preradical, we obtain the usual lattice of hereditary torsion classes.

Two applications of Nagata rings and modules

Let [Formula: see text] be a finite type hereditary torsion theory on the category of all modules over a commutative ring. The purpose of this paper is to give two applications of Nagata rings and modules in the sense of Jara [Nagata rings, Front. Math. China 10 (2015) 91–110]. First they are used to obtain Chase’s Theorem for [Formula: see text]-coherent rings. In particular, we obtain the [Formula: see text]-version of Chase’s Theorem, where [Formula: see text] is the classical star operation in ideal theory. In the second half, we apply they to characterize [Formula: see text]-flatness in the sense of Van Oystaeyen and Verschoren [Relative Invariants of Rings-The Commutative Theory, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 79 (Marcel Dekker, Inc., New York, 1983)].

On Differential Torsion Theories and Rings with Several Objects

AbstractLet${\mathcal{R}}$be a small preadditive category, viewed as a “ring with several objects.” A right${\mathcal{R}}$-module is an additive functor from${\mathcal{R}}^{\text{op}}$to the category$Ab$of abelian groups. We show that every hereditary torsion theory on the category$({\mathcal{R}}^{\text{op}},Ab)$of right${\mathcal{R}}$-modules must be differential.

Comultiplication modules relative to a hereditary torsion theory

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 properties and characterizations of τ-comultiplication modules. We also investigate modules for which every submodule has a unique τ-pseudo-complement and prove that every τ-comultiplication module is a module with unique τ-pseudo-complements.

Boyle’s conjecture and perfect localizations

ABSTRACTIn this article we study the behavior of left QI-rings under perfect localizations. We show that a perfect localization of a left QI-ring is a left QI-ring. We prove that Boyle’s conjecture is true for left QI-rings with finite Gabriel dimension such that every hereditary torsion theory in the Gabriel filtration is perfect. As corollary, we get that Boyle’s conjecture is true for left QI-rings which satisfy the restricted left socle condition, a result proved by Faith in [6].

A new class of simple noetherian V-domains

If R is any simple left noetherian, left hereditary, left V-domain, it is proven that the localization of R at any hereditary torsion theory τ that is cogenerated by a nonzero semisimple module, yields a ring of quotients Rτ with the same aforementioned properties. Examples of left V-domains R possessing (up to isomorphism) a single simple left R-module have been constructed by Cozzens (in 1970), and possessing infinitely many simple left R-modules, by Osofsky (in 1971). The methods developed in this paper can be used to construct V-domains possessing any prescribed number (finite or infinite) of simples. This answers in the affirmative a question posed by Cozzens and Faith in their book Simple Noetherian rings (Cambridge University Press, 1975).

Modular C11 lattices and lattice preradicals

This paper deals with properties of modular [Formula: see text] lattices involving hereditary preradicals on hereditary classes of modular lattices. Applications are given to Grothendieck categories and module categories equipped with hereditary torsion theories.

Singular and nonsingular modules relative to a torsion theory

ABSTRACTIn this paper, we introduce and study torsion-theoretic generalizations of singular and nonsingular modules by using the concept of τ-essential submodule for a hereditary torsion theory τ. We introduce two new module classes called τ-singular and non-τ-singular modules. We investigate some properties of these module classes and present some examples to show that these new module classes are different from singular and nonsingular modules. We give a characterization of τ-semisimple rings via non-τ-singular modules. We prove that if M∕τ(M) is non-τ-singular for a module M, then every submodule of M has a unique τ-closure. We give some properties of the torsion theory generated by the class of all τ-singular modules. We obtain a decomposition theorem for a strongly τ-extending module by using non-τ-singular modules.

HEREDITARY TORSION THEORIES OF A LOCALLY NOETHERIAN GROTHENDIECK CATEGORY

Let${\mathcal{A}}$be a locally noetherian Grothendieck category. We construct closure operators on the lattice of subcategories of${\mathcal{A}}$and the lattice of subsets of$\text{ASpec}\,{\mathcal{A}}$in terms of associated atoms. This establishes a one-to-one correspondence between hereditary torsion theories of${\mathcal{A}}$and closed subsets of$\text{ASpec}\,{\mathcal{A}}$. If${\mathcal{A}}$is locally stable, then the hereditary torsion theories can be studied locally. In this case, we show that the topological space$\text{ASpec}\,{\mathcal{A}}$is Alexandroff.

New results on C 11 and C 12 lattices with applications to Grothendieck categories and torsion theories

In this paper, which is a continuation of our previous paper [T. Albu, M. Iosif, A. Tercan, The conditions (C i ) in modular lattices, and applications, J. Algebra Appl. 15 (2016), http: dx.doi.org/10.1142/S0219498816500018], we investigate the latticial counterparts of some results about modules satisfying the conditions (C 11) or (C 12). Applications are given to Grothendieck categories and module categories equipped with hereditary torsion theories.