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1.
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   

2.
In this paper we present a new approach to Grothendieck duality over commutative rings. Our approach is based on the idea of rigid dualizing complexes, which was introduced by Van den Bergh in the context of noncommutative algebraic geometry. The method of rigidity was modified to work over general commutative base rings in our paper (Yekutieli and Zhang, Trans AMS 360:3211–3248, 2008). In the present paper we obtain many of the important local features of Grothendieck duality, yet manage to avoid lengthy and difficult compatibility verifications. Our results apply to essentially finite type algebras over a regular noetherian finite dimensional base ring, and hence are suitable for arithmetic rings. In the sequel paper (Yekutieli, Rigid dualizing complexes on schemes, in preparation) these results will be used to construct and study rigid dualizing complexes on schemes. This research was supported by the US–Israel Binational Science Foundation. The second author was partially supported by the US National Science Foundation.  相似文献   

3.
We survey the set–theoretic methods of module theory that make it possible to equip roots of the contravariant Ext functor with filtrations built from the small roots. The power of these methods is illustrated by several applications: a solution to the Kaplansky problem on Baer modules and some of the related problems for relative Baer modules, the structure of tilting modules and classes, the structure of Matlis localizations of commutative rings, and in particular cases, proofs of the finitistic dimension conjectures, and of the telescope conjecture for module categories. Received: January 2007  相似文献   

4.
The notion of differential Lie module over a curved coalgebra is introduced. The homotopy invariance of the structure of a differential Liemodule over a curved coalgebra is proved. A relationship between the homotopy theory of differential Lie modules over curved coalgebras and the theory of Koszul duality for quadratic-scalar algebras over commutative unital rings is determined.  相似文献   

5.
A submodule N of a module M is idempotent if N = Hom(M, N)N. The module M is fully idempotent if every submodule of M is idempotent. We prove that over a commutative ring, cyclic idempotent submodules of any module are direct summands. Counterexamples are given to show that this result is not true in general. It is shown that over commutative Noetherian rings, the fully idempotent modules are precisely the semisimple modules. We also show that the commutative rings over which every module is fully idempotent are exactly the semisimple rings. Idempotent submodules of free modules are characterized.  相似文献   

6.
We generalize results of Foxby concerning a commutative Nötherian ring to a certain noncommutative Nötherian algebra Λ over a commutative Gorenstein complete local ring. We assume that Λ is a Cohen–Macaulay isolated singularity having a dualizing module. Then the same method as in the commutative cases works and we obtain a category equivalence between two subcategories of mod Λ, one of which includes all finitely generated modules of finite Gorenstein dimension. We give examples of such algebras which are not Gorenstien; orders related to almost Bass orders and some k-Gorenstein algebras for an integer k.Presented by I. Reiten The author is supported by Grant-in-Aid for Scientific Researches B(1) No. 14340007 in Japan.  相似文献   

7.
Lourdes Juan  Andy Magid 《代数通讯》2013,41(10):4336-4346
Differential modules over a commutative differential ring which are projective as ring modules, with differential homomorphisms, form an additive category. Every projective ring module is shown occurs as the underlying module of a differential module. Differential modules, projective as ring modules, are shown to be direct summands of differential modules free as ring modules; those which are differential direct summands of differential direct sums of the ring being induced from the subring of constants. Every differential module finitely generated and projective as a ring module is shown to have this form after a faithfully flat finitely presented differential extension of the base.  相似文献   

8.
We study some non-highest weight modules over an affine Kac–Moody algebra [^(\mathfrak g)]{\hat{\mathfrak g}} at the non-critical level. Roughly speaking, these modules are non-commutative localizations of some non-highest weight “vacuum” modules. Using free field realization, we embed some rings of differential operators in endomorphism rings of our modules. These rings of differential operators act on a localization of the space of coinvariants of any [^(\mathfrak g)]{\hat{\mathfrak g}}-module with respect to a certain level subalgebra. In a particular case this action is identified with the Casimir connection.  相似文献   

9.
K. I. Beidar 《代数通讯》2013,41(11):4251-4258
In the present article we study the structure of rings, over which essential extensions of semisimple modules are direct sums of quasi-injectives. In the special case of commutative rings, these rings are precisely Artinian PIR and so every module over such rings is a direct sum of cyclics as characterized by Köthe and Cohen-Kaplansky.  相似文献   

10.
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