Torsion Functor Research Articles

Exact BCK-sequence in BCK-algebra

The notion of quasi BCK-exact sequence of modules was introduced by B.Davvaz and coauthors in 1999 as a generalization of the notion of exact sequence. In 2016 S. Asawasamrit & C. Promsakon introduced Quasi-Commutative KK-algebra [14]. In this paper we investigate further this notion on BCK-algebra. In particular, some interesting results concerning this concept and torsion functor on BCK-algebra are given.

The locally nilradical for modules over commutative rings

Let $R$ be a commutative unital ring and $a\in R.$ We introduce and study properties of a functor $a\Gamma_{a}(-),$ called the locally nilradical on the category of $R$-modules. $a\Gamma_{a}(-)$ is a generalisation of both the torsion functor (also called section functor) and Baer's lower nilradical for modules. Several local-global properties of the functor $a\Gamma_{a}(-)$ are established. As an application, results about reduced $R$-modules are obtained and hitherto unknown ring theoretic radicals as well as structural theorems are deduced.

Injective stabilization of additive functors. II. (Co)torsion and the Auslander-Gruson-Jensen functor

The formalism of injective stabilization of additive functors is used to define a new notion of the torsion submodule of a module. It applies to arbitrary modules over arbitrary rings. For arbitrary modules over commutative domains it coincides with the classical torsion, and for finitely presented modules over arbitrary rings it coincides with the Bass torsion. A formally dual approach – based on projective stabilization – gives rise to a new concept: the cotorsion quotient module of a module. This is done in complete generality – the new concept is defined for any module over any ring. Unlike torsion, cotorsion does not have classical prototypes.General properties of these constructs are established. It is shown that the Auslander-Gruson-Jensen functor applied to the cotorsion functor returns the torsion functor. As a consequence, a ring is one-sided absolutely pure if and only if each pure injective on the other side is cotorsion-free. If the injective envelope of the ring is finitely presented, then the right adjoint of the Auslander-Gruson-Jensen functor applied to the torsion functor returns the cotorsion functor. This correspondence establishes a duality between torsion and cotorsion over such rings. In particular, this duality applies to artin algebras. It is also shown that, over any ring, the character module of the torsion of a module is isomorphic to the cotorsion of the character module of the module. Under various finiteness conditions on the injective envelope of the ring, the derived functors of torsion and cotorsion are computed.

Weak proregularity, weak stability, and the noncommutative MGM equivalence

Let A be a commutative ring, and let a be a finitely generated ideal in it. It is known that a necessary and sufficient condition for the derived a-torsion and a-adic completion functors to be nicely behaved is the weak proregularity of a. In particular, the MGM Equivalence holds.Because weak proregularity is defined in terms of elements of the ring (it involves limits of Koszul complexes), it is not suitable for noncommutative ring theory.In this paper we introduce a new condition on a torsion class T in a module category: weak stability. Our first main theorem says that in the commutative case, the ideal a is weakly proregular if and only if the corresponding torsion class T is weakly stable.We then study weak stability of torsion classes in module categories over noncommutative rings. There are three main theorems in this context: ▷ For a torsion class T that is weakly stable, quasi-compact and finite dimensional, the right derived torsion functor is isomorphic to a left derived tensor functor. ▷ The Noncommutative MGM Equivalence, that holds under the same assumptions on T. ▷ A theorem about symmetric derived torsion for complexes of bimodules. This last theorem is a generalization of a result of Van den Bergh from 1997, and corrects an error in a paper of Yekutieli & Zhang from 2003.

Assassins and Torsion Functors

Let $R$ be a ring, let $\mathfrak{a}\subseteq R$ be an ideal, and let $M$ be an $R$-module. Let $\Gamma_{\mathfrak{a}}$ denote the $\mathfrak{a}$-torsion functor. Conditions are given for the (weakly) associated primes of $\Gamma_{\mathfrak{a}}(M)$ to be the (weakly) associated primes of $M$ containing $\mathfrak{a}$, and for the (weakly) associated primes of $M/\Gamma_{\mathfrak{a}}(M)$ to be the (weakly) associated primes of $M$ not containing $\mathfrak{a}$.

On the degree of Hilbert polynomials associated to the torsion functor

This is the published version, also available here: http://dx.doi.org/10.1090/S0002-9939-07-08879-X. First published in Proc. Amer. Math. Soc. in 2009, published by the American Mathematical Society.

DECOMPOSITIONS OF MODULES SUPPLEMENTED RELATIVE TO A TORSION THEORY

Let R be a ring, M a right R-module and a hereditary torsion theory in Mod-R with associated torsion functor τ for the ring R. Then M is called τ-supplemented when for every submodule N of M there exists a direct summand K of M such that K ≤ N and N/K is τ-torsion module. In [4], M is called almost τ-torsion if every proper submodule of M is τ-torsion. We present here some properties of these classes of modules and look for answers to the following questions posed by the referee of the paper [4]: (1) Let a module M = M′ ⊕ M″ be a direct sum of a semisimple module M′ and τ-supplemented module M″. Is M τ-supplemented? (2) Can one find a non-stable hereditary torsion theory τ and τ-supplemented modules M′ and M″ such that M′ ⊕ M″ is not τ-supplemented? (3) Can one find a stable hereditary torsion theory τ and a τ-supplemented module M such that M/N is not τ-supplemented for some submodule N of M? (4) Let τ be a non-stable hereditary torsion theory and the module M be a finite direct sum of almost τ-torsion submodules. Is M τ-supplemented? (5) Do you know an example of a torsion theory τ and a τ-supplemented module M with τ-torsion submodule τ(M) such that M/τ(M) is not semisimple?

Bivariate Hilbert functions for the torsion functor

Let ( R, P) be a commutative, local Noetherian ring, I, J ideals, M and N finitely generated R-modules. Suppose J+ ann R M+ ann R N is P-primary. The main result of this paper is Theorem 6, which gives necessary and sufficient conditions for the length of Tor i ( M/ I n M, N/ J m N), to agree with a polynomial, for m, n⪢0. As a corollary, it is shown that the length of Tor i ( M/ I n M, N/ I n N) always agrees with a polynomial in n, for n⪢0, provided I+ ann R M+ ann R N is P-primary.

The cohomology of a pull-back on [formula omitted]-formal spaces

We prove a collapse theorem for the Eilenberg–Moore spectral sequence with coefficients in a field K converging to the cohomology of the pull-back of a fibration q :E→B by a continuous map f :X→B when E, X and B are K -formal. We also show that the cohomology algebra of the pull-back can be expressed via the torsion functor with the shc-minimal model for B in the sense of Ndombol and Thomas [Topology 41 (2002) 85] and its free extensions for E and X without the assumption of K -formality. Moreover not only does the shc-minimal models for E, B and X enable us to construct a model for the Eilenberg–Moore spectral sequence also they help in computing the spectral sequence.

The smallest closed subcategory containing the μcomplemented modules

Given a hereditary torsion functor , the class of μ-complemented modules was recently introduced by P.F. Smith and the authors as an analogue of extending modules. This current article explores this class by viewing it as a subclass of EX, the smallest closed subcategory which contains it. As a consequence the class of μ-complemented modules is shown here to be closed under the formation of module of quotients. As to EX, we prove that it is closed under arbitrary direct products in Mod-Rif Ris a valuation ring. On the other hand, if Ris commutative Noetherian and μ is jansian then every μ-complemented module is a direct sum of a μ-torsion module and a semisimple, which prompted us to analyze when E Xcontains a subgenerator of this form.

Homology with Models and Tor

We consider the extension of the notion of a projective module to that of a projective functor relative to a model set (as in Dold, MacLane, Oberst, 1967). Then taking projective resolutions of functors, we consider the usual associated homology. We show that in some cases, including the classical simplicial homology of topological spaces, the model set can be replaced by a model set having only one element. We show that when the model set consists of a single element the homology modules can be interpreted as values of the Torsion functor. In the case of simplicial homology of topological spaces these Tors will be shown to be analogous to the Tors which occur in group homology.

Modules complemented with respect to a torsion theory

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.

On μ-complemented and singular modules

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.

Derived functors of the torsion functor and local cohomology of noncommutative rings

AbstractLet R be an associative ring which is not necessarily commutative. For any torsion theory τ on the category of left R-modules and for any nonnegative integer n we define and study the notion of the nth local cohomology functor with respect to τ. For suitably nice rings a bound for the nonvanishing of these functors is given in terms of the τ-dimension of the modules.

On the associativity of the torsion functor

Let R be a commutative ring with identity. We say that tor is associative over R if for all R-modules A, B, C there is an isomorphism Our main results are that (1) tor is associative over a noetherian ring R if and only if R is the direct sum of a finite number of Dedekind rings and uniserial rings, and (2) tor is associative over an integral domain R if and only if R is a Prüfer ring.