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1.
Weimin Xue 《代数通讯》2013,41(7):2243-2247
A module over an artinian ring is uniserial if it has a unique composition series, and an artinian ring is serial if each of its indecomposable projective modules is uniserial. Fuller [4, Theorem 5.4] showed that an artinian ring R is serial if and only if each of left indecomposable projective and injective R-modules is uniserial. The following question was raised in 4, p.134: Is an artinian ring R necessarily serial if each of its indecomposable injective modules is uniserial? Example 1 in this note answers this question in the negative 相似文献
2.
《代数通讯》2013,41(11):4265-4271
Abstract In this paper we consider a generalization of Vámos's proposition which asserts that finitely generated artinian and distributive modules are cyclic. Also, we show its duality as an answer to the problem which is shown by him (See Vámos, [Vámos, P. (1978). Finitely generated artinian and distributive modules are cyclic. Bull. London Math. Soc. 10(3):287–288]). 相似文献
3.
称左R-模M是ecg-扩张模,如果M的任意基本可数生成子模是M的直和因子的基本子模.在研究了ecg-扩张模的基本性质的基础上,本文证明了对于非奇异环R,所有左R-模是ecg-扩张模当且仅当所有左R-模是扩张模.同时我们还用ecg-拟连续模刻画了Noether环和Artin半单环. 相似文献
4.
It is proved that A is a right distributive ring if and only if all quasiinjective right A-modules are Bezout left modules over their endomorphism rings if and only if for any quasiinjective right A-module M which is a Bezout left End (M)-module, every direct summand N of M is a Bezout left End(N)-module. If A is a right or left perfect ring, then all right A-modules are Bezout left modules over their endomorphism rings if and only if all right A-modules are distributive left modules over their endomorphism rings if and only if A is a distributive ring. 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.
7.
A result of Ginn and Moss asserts that a left and right noetherian ring with essential right socle is left and right artinian. There are examples of right finitely embedded rings with ACC on left and right annihilators which are not artinian. Motivated by this, it was shown by Faith that a commutative, finitely embedded ring with ACC on annihilators (and square-free socle) is artinian (quasi-Frobenius). A ring R is called right minsymmetric if, whenever k R is a simple right ideal of R, then R k is also simple. In this paper we show that a right noetherian right minsymmetric ring with essential right socle is right artinian. As a consequence we show that a ring is quasi-Frobenius if and only if it is a right and left mininjective, right finitely embedded ring with ACC on right annihilators. This extends the known work in the artinian case, and also extends Faith's result to the non-commutative case. 相似文献
8.
Harlan L. Hullinger 《Journal of Pure and Applied Algebra》1980,16(3):265-273
The representation theory of a ring Δ has been studied by examining the category of contravariant (additive) functors from the category of finitely generated left Δ-modules to the category of abelian groups. Closely connected with the representation theory of a ring is the study of stable equivalence, which is a relaxing of the notion of Morita equivalence. Here we relate two stably equivalent rings via their respective functor categories and examine left artinian rings with the property that every left Δ-module is a direct sum of finitely generated modules. 相似文献
9.
An associative ring R, not necessarily with an identity element, is called semilocal if R modulo its Jacobson radical is an artinian ring. It is proved that if the adjoint group of a semilocal ring R is locally supersoluble, then R is locally Lie-supersoluble and its Jacobson radical is contained in a locally Lie-nilpotent ideal of finite index in R. 相似文献
10.
While a module is pseudo-injective if and only if it is automorphism-invariant, it was not known whether automorphism-invariant modules are tight. It is shown that weakly automorphism-invariant modules are precisely essentially tight. We give various examples of weakly automorphism-invariant and essentially tight modules and study their properties. Some particular results: (1) R is a semiprime right and left Goldie ring if and only if every right (left) ideal is weakly injective if and only if every right (left) ideal is weakly automorphism invariant; (2) R is a CEP-ring if and only if R is right artinian and every indecomposable projective right R-module is uniform and essentially R-tight. 相似文献
11.
S. Jondrup J. Krempa D. Niewieczerzal 《Rendiconti del Circolo Matematico di Palermo》1988,37(1):100-108
In this note we study rings having only a finite number of non isomorphic uniform modules with non zero socle. It is proved
that a commutative ring with this property is a direct sum of a finite ring and a ring of finite representation type. In the
non commutative case we show that most P.I. rings having only a finite number of non isomorphic modules with non zero socle
are in fact artinian. 相似文献
12.
It is proved that a semiperfect module is lifting if and only if it has a projective cover preserving direct summands. Three corollaries are obtained: (1) every cyclic module over a ring R is lifting if and only if every cyclic R-module has a projective cover preserving direct summands; (2) a ring R is artinian serial with Jacobson radical square-zero if and only if every (2-generated) R-module has a projective cover preserving direct summands; (3) a ring R is a right (semi-)perfect ring if and only if (cyclic) lifting R-module has a projective cover preserving direct summands, if and only if every (cyclic) R-module having a projective cover preserving direct summands is lifting. It is also proved that every cyclic module over a ring R is ⊕-supplemented if and only if every cyclic R-module is a direct sum of local modules. Consequently, a ring R is artinian serial if and only if every left and right R-module is a direct sum of local modules. 相似文献
13.
本文用则模的术语给出了半单Artin 环的刻划。得到如下三个条件的等价性:(1)R 是一个半单Artin 环;(2)每一个R-模都是正则模;(3)每一个单纯R-模都是正则模。 相似文献
14.
On Graded Distributive Modules 总被引:1,自引:0,他引:1
Chen Qinghua 《东北数学》1994,(2)
OnGradedDistributiveModules¥ChenQinghua(陈清华)(DepartmentofMathematics,FujianNormalUniversity,Fuzhou,350007)Abstract:Inthispape... 相似文献
15.
In this paper, we define and study weakly distributive modules as a proper generalization of distributive modules.We prove
that, weakly distributive supplemented modules are amply supplemented. In a weakly distributive supplemented module every
submodule has a unique coclosure. This generalizes a result of Ganesan and Vanaja. We prove that π-projective duo modules, in particular commutative rings, are weakly distributive. Using this result we obtain that in a commutative
ring supplements are unique. This generalizes a result of Camillo and Lima. We also prove that any weakly distributive ⊕-supplemented
module is quasi-discrete. 相似文献
16.
A. A. Tuganbaev 《Mathematical Notes》1996,60(2):186-203
A module is called distributive (is said to be a chain module) if the lattice of all its submodules is distributive (is a chain). Let a ringA be a finitely generated module over its unitary central subringR. We prove the equivalence of the following conditions:
- A is a right or left distributive semiprime ring;
- for any maximal idealM of a subringR central inA, the ring of quotientsA M is a finite direct product of semihereditary Bézout domains whose quotient rings by the Jacobson radicals are finite direct products of skew fields;
- all right ideals and all left ideals of the ringA are flat (right and left) modules over the ringA, andA is a distributive ring, without nonzero nilpotent elements, all of whose quotient rings by prime ideals are semihereditary orders in skew fields.
17.
18.
AbstractWe say that a class Q of left R-modules is a monic class if a nonzero submodule of a module in Q is also a module in Q. For a monic class Q, we define a Q-dimension of modules that measures how far modules are from the modules in Q. For a monic class Q of indecomposable modules we characterize rings whose modules have Q-dimension. We prove that for an artinian principal ideal ring the Q-dimension coincides with the uniserial dimension. We also characterize when every module has Q-dimension. 相似文献
19.
Nguyen Viet dung 《代数通讯》2013,41(8):3921-3947
Let R be a left artinian ring such that every finitely presented right .ft-module is of finite endolength. It is shown that the cardinality of the set of isomorphism classes of preinjective right R-modules is less than or equal to the cardinality of the set of isomorphism classes of preprojective left R-modules, and R is of finite representation type if and only if these cardinal numbers are finite and equal to each other. As a consequence, we deduce a theorem, due to Herzog [17], asserting that a left pure semisimple ring R is of finite representation type if and only if the number of non-isomorphic preinjective right R-modules is the same as the number of non-isomorphic preprojective left .R-modules. Further applications are also given to provide new criteria for artinian rings with self-duality and artinian Pi-rings to be of finite representation type, which imply in particular the validity of the pure semisimple conjecture for these classes of rings. 相似文献
20.
《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