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1.
Ladislav Bican 《Czechoslovak Mathematical Journal》2008,58(2):561-568
In the class of all exact torsion theories the torsionfree classes are cover (pre-cover) classes if and only if the classes
of torsionfree relatively injective modules or relatively exact modules are cover (precover) classes, and this happens exactly
if and only if the torsion theory is of finite type. Using the transfinite induction in the second half of the paper a new
construction of a torsionfree relatively injective cover of an arbitrary module with respect to Goldie’s torsion theory of
finite type is presented.
This research has been partially supported by the Grant Agency of the Czech Republic, grant #GAČR 201/06/0510 and also by
the institutional grant MSM 0021620839. 相似文献
2.
We consider R‐torsionfree modules over group rings RG, where R is a Dedekind domain and G is a finite group. In the first part of the paper [4] we compared the theory T(RG) of all R‐torsionfree RG‐modules and the theory T0(RG) of RG‐lattices (i. e. finitely generated R‐torsionfree RG‐modules), and we realized that they are almost always different. Now we compare their behaviour with respect to decidability, when RG‐lattices are of finite, or wild representation type. 相似文献
3.
In this article, we first study the existence of envelopes and covers by modules of finite divisible and torsionfree dimensions. Then we investigate divisible and torsionfree dimensions as well as localizations of divisible and torsionfree modules over commutative rings. Finally, Gorenstein divisible and torsionfree modules are introduced and studied. 相似文献
4.
We consider R‐torsionfree modules over group rings RG, where R is a Dedekind domain and G is a finite group. We compare the (first order) theory T of al these modules and the theory T0 of the finitely generated ones (so of RG‐lattices). It is easy to realize that they are equal iff R is a field. The obstruction is the existence of R‐divisible R‐torsionfree RG‐modules. Accordingly we consider R‐reduced R‐torsionfree RG‐modules for a local R. We show that the key conditions ensuring that their theory equals T0 are: (1) RG‐lattices have a finite representation type; (2) each attice over the completion R̂G is isomorphic to the completion of some RG‐lattice.Some related questions are discussed. 相似文献
5.
A well-known result of Small states that if M is a noetherian left R-module having endomorphism ring S then any nil subring of S is nilpotent. Fisher [4] dualized this result and showed that if M is left artinian then any nil ideal of S is nilpotent. He gave a bound on the indices of nilpotency of nil subrings of the endomorphism rings of noetherian modules and raised the dual question of whether there are such bounds in the case of artinian modules. He gave an affirmative answer if the module is also assumed to be finitely-generated. Similar affirmative answers for modules with finite homogeneous length were given in [10] and [15]. On the other hand, the nilpotence of certain ideals of the endomorphism rings of modules noetherian relative to a torsion theory has been extensively studied. See [2,6,8,12,15,17]. Jirasko [11] dualized, in some sense, some of the results of [6] to torsion modules satisfying the descending chain conditions with respect to some radical. In this paper we give a bound of indices of nilpotency on nil subrings of the endomorphism ring of a left R-module which is T-torsionfree with respect to some torsion theory T on R-mod. As a special case, we obtain an affirmative answer to Fisher's question. We also note that our results can be stated in an arbitrary Grothendieck category. 相似文献
6.
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. 相似文献
7.
Pascual Jara 《Czechoslovak Mathematical Journal》2007,57(1):225-241
Lattices of submodules of modules and the operators we can define on these lattices are useful tools in the study of rings
and modules and their properties. Here we shall consider some submodule operators defined by sets of left ideals. First we
focus our attention on the relationship between properties of a set of ideals and properties of a submodule operator it defines.
Our second goal will be to apply these results to the study of the structure of certain classes of rings and modules. In particular
some applications to the study and the structure theory of torsion modules are provided. 相似文献
8.
This work presents a new construction of algebraic completion using techniques from torsion theory and including the classical car,e of a-completion as an example. The structure of the completed rings and modules and related problems are studied in detail. 相似文献
9.
Jonathan S. Golan 《Israel Journal of Mathematics》1983,45(2-3):257-264
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. 相似文献
10.
Lidia Angeleri Hügel Alberto Tonolo Jan Trlifaj 《Algebras and Representation Theory》2001,4(2):155-170
We relate the theory of envelopes and covers to tilting and cotilting theory, for (infinitely generated) modules over arbitrary rings. Our main result characterizes tilting torsion classes as the pretorsion classes providing special preenvelopes for all modules. A dual characterization is proved for cotilting torsion-free classes using the new notion of a cofinendo module. We also construct unique representing modules for these classes. 相似文献
11.
Richard Wiegandt 《Semigroup Forum》2006,72(2):312-328
After recapitulating the rudiments of the Kurosh-Amitsur
radical theory of S-acts, hereditary radicals are discussed.
The hereditary Hoehnke radical assignments r which designate a
Rees congruence r(A) to each S-act A, are the hereditary
Kurosh-Amitsur radical assignments.
Then the corresponding radical as well as semisimple classes are
characterized.
Equivalence classes of injective S-acts determine hereditary
torsion assignments t, these are just the hereditary Hoehnke
radicals, but the congruence t(A) of A need not be a Rees
congruence.
Torsion and torsionfree classes are characterized; several
hereditary torsion assignments may determine the same torsion
class which is always a radical class closed under taking subacts.
Examples show that a hereditary torsion assignment need not be a
hereditary radical. 相似文献
12.
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. 相似文献
13.
T拟内射模与TQI环 总被引:2,自引:0,他引:2
本文定义并刻划了T拟内射模与T拟内射包,证明了WG-cocriticalT拟内射模的自同态环为正则非奇异右自内射环.最后讨论了T拟内射模与T内射模一致的环,即TQI环.还给出了Gabriel拓扑G中每个右理想T拟内射的几个等价条件 相似文献
14.
The study of pairs of modules (over a Dedekind domain) arises from two different perspectives, as a starting step in the analysis
of tuples of submodules of a given module, or also as a particular case in the analysis of Abelian structures made by two
modules and a morphism between them. We discuss how these two perspectives converge to pairs of modules, and we follow the
latter one to obtain an alternative approach to the classification of pairs of torsionfree objects. Then we restrict our attention
to pairs of free modules. Our main results are that the theory of pairs of free Abelian groups is co-recursively enumerable,
and that a few remarkable extensions of this theory are decidable.
Work performed under the auspices of GNSAGA-INDAM 相似文献
15.
《Quaestiones Mathematicae》2013,36(3):465-474
Abstract This paper surveys a selection of results in the literature on torsion preradicals; these are left exact preradical functors on the category of unital right modules over an associative ring with identity. Various well known classes of rings such as semisimple, artinian, perfect and strongly prime are characterized in terms of torsion preradicals. A classification of prime rings using torsion preradicals is also exhibited. Rings all of whose torsion preradicals are radicals and rings whose torsion preradicals commute, are investigated. An application of the latter condition to Jacobson's Conjecture is presented. 相似文献
16.
This paper studies the existence and properties of a torsion-free cover with respect to a faithful hereditary torsion theory
(T, F) of modules over a ring with unity. A direct sum of a finite number of torsion-free covers of modules is the torsion-free
cover of the direct sum of the modules. The concept of aT-near homomorphism, which generalizes Enochs’ definition of a neat submodule, is introduced and studied. This allows the generalization
of a result of Enochs on liftings of homomorphisms. Hereditary torsion theories for which every module has a torsion-free
cover are called universally covering. If the inclusion map ofR into the appropriate quotient ringQ is a left localization in the sense of Silver, the problem of the existence of universally-covering torsion theories can
be reduced to the caseR=Q. As a consequence, many sufficient conditions for a hereditary torsion theory to be universally covering are obtained. For
a universally-covering hereditary torsion theory (T, F), the following conditions are equivalent: (1) the product ofF-neat homomorphisms is alwaysT-neat; (2) the product of torsion-free covers is alwaysT-neat; (3) every nonzero module inT has a nonzero socle. 相似文献
17.
Auslander-Reiten Triangles, Ziegler Spectra and Gorenstein Rings 总被引:1,自引:0,他引:1
Apostolos Beligiannis 《K-Theory》2004,32(1):1-82
We investigate (existence of) Auslander—Reiten triangles in a triangulated category in connection with torsion pairs, existence of Serre functors, representability of homological functors and realizability of injective modules. We also develop an Auslander—Reiten theory in a compactly generated triangulated category and we study the connections with the naturally associated Ziegler spectrum. Our analysis is based on the relative homological theory of purity and Brown's Representability Theorem. Our main interest lies in the structure of Auslander—Reiten triangles in the full subcategory of compact objects. We also study the connections and the interplay between Auslander—Reiten theory, pure-semisimplicity and the finite type property, Grothendieck groups, and we give applications to derived categories of Gorenstein rings. 相似文献
18.
19.
Indah Emilia Wijayanti 《代数通讯》2013,41(4):1308-1333
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. 相似文献
20.
An analog of the reduction theorem for modules over an integrally closed integral domain is proved for torsionfree modules over a semiprimary Noetherian algebra. The result is translated into concrete terms for pseudo-Bass and pseudohereditary algebras, and also used to study the structure of genera.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 5, pp. 604–610, May, 1990. 相似文献