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In this paper, which is a continuation of our previous paper [T. Albu, M. Iosif, A. Tercan, The conditions ( C) 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) or ( C). Applications are given to Grothendieck categories and module categories equipped with hereditary torsion theories. [ABSTRACT FROM AUTHOR]

The purpose of this article is to prove that the category of cocommutative Hopf K-algebras, over a field K of characteristic zero, is a semi-abelian category. Moreover, we show that this category is action representable, and that it contains a torsion theory whose torsion-free and torsion parts are given by the category of groups and by the category of Lie K-algebras, respectively. [ABSTRACT FROM AUTHOR]

A semi-localization of a category is a full reflective subcategory with the property that the reflector is semi-left-exact. There are many interesting examples of semi-localizations, as for instance any torsion-free subcategory of a semi-abelian category. By specializing a result due to S. Mantovani, we first characterize the categories which are semi-localizations of exact Mal'tsev categories. We then prove a new characterization of protomodular categories in terms of binary relations, allowing us to obtain an abstract characterization of the semi-localizations of exact protomodular categories. This result is very useful to study the (hereditarily)-torsion-free subcategories of semi-abelian categories. Some examples are considered in detail in the categories of groups, crossed modules, commutative rings and topological groups. We finally explain how these results extend the corresponding ones obtained in the abelian context by W. Rump. [ABSTRACT FROM AUTHOR]

We characterize t-structures in stable ∞-categories as suitable quasicategorical factorization systems. More precisely we show that a t-structure 픱 on a stable ∞-category C is equivalent to a normal torsion theory 픽 on C, i.e. to a factorization system 픽 = (퓔, ℳ) where both classes satisfy the 3-for-2 cancellation property, and a certain compatibility with pullbacks/pushouts. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper we introduce and investigate the latticial counterpart of the module-theoretical concept of preradical. Applications are given to Grothendieck categories and module categories equipped with hereditary torsion theories. [ABSTRACT FROM AUTHOR]

In this paper, we study module theoretic definitions of the Baer and related ring concepts. We say a module is s.Baer if the right annihilator of a nonempty subset of the module is generated by an idempotent in the ring. We show that s.Baer modules satisfy a number of closure properties. Under certain conditions, a torsion theory is established for the s.Baer modules, and we provide examples of s.Baer torsion modules and modules with a nonzero s.Baer radical. The other principal interest of this paper is to provide explicit connections between s.Baer modules and projective modules. Among other results, we show that every s.Baer module is an essential extension of a projective module. Additionally, we prove, with limited and natural assumptions, that in a generalized triangular matrix ring every s.Baer submodule of the ring is projective. As an application, we show that every prime ring with a minimal right ideal has the strong summand intersection property. Numerous examples are provided to illustrate, motivate, and delimit the theory. [ABSTRACT FROM AUTHOR]

We first characterize τ-complemented modules with relative (pre)covers. We also introduce an extending module relative to τ-pure submodules on a hereditary torsion theory τ and give its relationship with τ-complemented modules. [ABSTRACT FROM AUTHOR]

Using the concept of prime submodule forM ∈ R-Mod,P ∈ Spec(M), andN ∈ σ[M], we define whenNisP-Mtame\ (Mtame) module. This concept generalizes the concept\ ofP-tame(tame) modules. ForM ∈ R-Mod and τ ∈M-tors, we use the concept of τM-Gabriel dimension and we study the relationship betweenMtamemodules and τM-Gabriel dimension. We find equivalent conditions for a moduleMprogenerator in σ[M] with τM-Gabriel dimension to have τM-Gabriel correspondence in terms of theP-Mtamemodules. This result extends the results by Albu et al. and Kim and Krause. [ABSTRACT FROM AUTHOR]

We formally introduce and study generalized comatrix coalgebras and upper triangular comatrix coalgebras, which are not only a dualization but also an extension of classical generalized matrix algebras. We use these to answer several open questions on Noetherian and Artinian type notions in the theory of coalgebras, and to give complete connections between these. We also solve completely the so called finite splitting problem for coalgebras: we show thatis a coalgebra such that the rational part of every left finitely generated-module splits off if and only ifis an upper triangular matrix coalgebra, for a serial coalgebrawhose Ext-quiver is a finite union of cycles, a finite dimensional coalgebraand a finite dimensional bicomodule. [ABSTRACT FROM AUTHOR]

Using the nice properties of thew-divisible weight and thew-divisible groups, we prove a factorization theorem for compact abelian groupsK; namely,K = Ktor × Kd, whereKtoris a bounded torsion compact abelian group andKdis aw-divisible compact abelian group. By Pontryagin duality this result is equivalent to the same factorization for discrete abelian groups proved in [9]. [ABSTRACT FROM AUTHOR]

We give a method to study the finiteness of the coradical filtration of a coalgebra; as a consequence, we show that a left semiartinian profinite algebra has nilpotent Jacobson radical and is right semiartinian too. Equivalently, we show that a for a semilocal profinite algebra, T-nilpotence implies nilpotence for the Jacobson radical. This answers two open questions from Iovanov et al. (J Algebra 320(5):2144-2155, ). [ABSTRACT FROM AUTHOR]

Bezrukavnikov, later together with Arinkin, recovered Deligne's work defining perverse t-structures in the derived category of coherent sheaves on a projective scheme. We prove that these t-structures can be obtained through tilting with respect to torsion theories, as in the work of Happel, Reiten and Smalø. This approach allows us to define, in the quasi-coherent setting, similar perverse t-structures for certain noncommutative projective planes. [ABSTRACT FROM AUTHOR]

Using the concept of prime submodule defined by Raggi et al. in [16], forM ∈ R-Mod we define the concept of classical Krull dimension relative to a hereditary torsion theory τ ∈M-tors. We prove that ifMis progenerator in σ[M], τ ∈M-tors such thatMhas τ-Krull dimension thencl.Kτdim (M) ≤ kτ(M). Also we show that ifMis noetherian, τ-fully bounded, progenerator of σ[M], andM∈ 𝔽τ, thencl·Kτdim (M) = kτ(M). [ABSTRACT FROM AUTHOR]

Abstract: Bijective correspondences are established between endofinite injective left modules, endofinite flat right modules, finite collections of minimal noetherian prime ideals, normalized rank functions on left ideals and characters. Endofinite flat modules are identified as flat covers of modules associated to a minimal noetherian prime ideal, while endofinite flat injectives are characterized by localizations with a semiprimary QF-3 quotient ring. [Copyright &y& Elsevier]

This is the second part of the paper with the same title published inCommunications in Algebrain 2011. It contains applications of the Latticial Osofsky–Smith Theorem to Grothendieck categories and module categories equipped with a torsion theory. Various many different meanings spread in the literature of the relative concepts with respect to a hereditary torsion theory τ on Mod-Rlike τ-essential submodule, τ-complement submodule, τ-CS module, etc. are also discussed. [ABSTRACT FROM PUBLISHER]

We classify the torsion pairs in a tube category and show that they are in bijection with maximal rigid objects in the extension of the tube category containing the Prüfer and adic modules. We show that the annulus geometric model for the tube category can be extended to the larger category and interpret torsion pairs, maximal rigid objects and the bijection between them geometrically. We also give a similar geometric description in the case of the linear orientation of a Dynkin quiver of type A. [ABSTRACT FROM AUTHOR]

We show that every two-term tilting complex over an Artin algebra has a tilting module over a certain factor algebra as a homology group. Also, we determine the endomorphism algebra of such a homology group, which is given as a certain factor algebra of the endomorphism algebra of the two-term tilting complex. Thus, every derived equivalence between Artin algebras given by a two-term tilting complex induces a derived equivalence between the corresponding factor algebras. [ABSTRACT FROM PUBLISHER]

Abstract: Given a torsion theory in an abelian category , the reflector to the torsion-free subcategory induces a reflective factorisation system on . It was shown by A. Carboni, G.M. Kelly, G. Janelidze and R. Paré that induces a monotone-light factorisation system by simultaneously stabilising and localising , whenever the torsion theory is hereditary and any object in is a quotient of an object in . We extend this result to arbitrary normal categories, and improve it also in the abelian case, where the heredity assumption on the torsion theory turns out to be redundant. Several new examples of torsion theories where this result applies are then considered in the categories of abelian groups, groups, topological groups, commutative rings, and crossed modules. [Copyright &y& Elsevier]

In this article we define and compare different types of the notion of grade with respect to torsion theories over commutative rings which are not necessarily Noetherian. We do this by using Ext-modules, Koszul cohomology modules, and Čech and local cohomology modules. An application of these results is given. [ABSTRACT FROM PUBLISHER]

We study the existence of covers and envelopes by some special functors on the category of finitely presented modules. As an application, we characterize some important rings using these functors. We also investigate homological properties of some functors on the stable module category. The relationship between phantoms and Ext-phantoms is obtained. It is shown that every leftR-moduleMhas an Ext-phantom preenvelopef:M → Nwith coker(f) pure-projective. Finally, we prove that, as a torsionfree class of (mod-R, Ab), (mod-R, Ab) is generated by theFP-injective objects. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM PUBLISHER]

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. [ABSTRACT FROM AUTHOR]

Let R be a ring. A collection of R-modules containing the zero module and closed under isomorphisms will be denoted by X. An R-module M is said to be X -lifting if for every X-submodule N of M there exists A ≤ N such that M = A ⊕ B and N ∩ B is small in B [11]. In the present paper, we consider the question: Can we characterize X-lifting modules via objects of the class X ? [ABSTRACT FROM AUTHOR]

Let R be a ring and let A be a hereditary torsion class of R-modules. The inclusion of the localizing subcategory generated by A into the derived category of R has a right adjoint, denoted CellA. Recently, Benson has shown how to compute CellA R when R is a group ring of a finite group over a prime field and A is the hereditary torsion class generated by a simple module. We generalize Benson's construction to the case where A is any hereditary torsion class on R. It is shown that for every R-module M there exists an injective R-module E such that: Hn (CellAM) ≅ Ext Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. (Hom R(M,E),E) for n ≥ 2. [ABSTRACT FROM AUTHOR]

We consider a generalization of lifting modules relative to a class $\mathcal{A}$ of modules and a proper class 피 of short exact sequences of modules. These modules will be called 피-$\mathcal{A}$-lifting. We establish characterizations of modules with the property that every direct sum of copies of them is 피-$\mathcal{A}$-lifting. [ABSTRACT FROM AUTHOR]

An R-module M is said to be a C1-module if every closed submodule of M is a direct summand. In this paper we introduce and investigate the concept of the τ-C1 module for a hereditary torsion theory τ on Mod-R. τ -C1 modules are a generalization of C1-modules. [ABSTRACT FROM AUTHOR]

Let R be a ring. A right R-module U is called Tor-tilting if [image omitted], where U+ = Hom(U, /), Cogen(U+) is the class of left R-modules cogenerated by U+ and [image omitted] consists of modules RM such that [image omitted]. Some examples and characterizations of Tor-tilting modules are given. Among others, it is shown that UR 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. [ABSTRACT FROM AUTHOR]