On weakly prime radical of modules and semi-compatible modules

Summary Let M be a left R-module. Then a proper submodule P of M is called weakly prime submodule if for any ideals A and B of R and any submodule N of M such that ABN ⊆ P, we have AN ⊆ P or BN ⊆ P. We define weakly prime radicals of modules and show that for Ore domains, the study of weakly prime radicals of general modules reduces to that of torsion modules. We determine the weakly prime radical of any module over a commutative domain R with dim (R) ≦ 1. Also, we show that over a commutative domain R with dim (R) ≦ 1, every semiprime submodule of any module is an intersection of weakly prime submodules. Localization of a module over a commutative ring preserves the weakly prime property. An R-module M is called semi-compatible if every weakly prime submodule of M is an intersection of prime submodules. Also, a ring R is called semi-compatible if every R-module is semi-compatible. It is shown that any projective module over a commutative ring is semi-compatible and that a commutative Noetherian ring R is semi-compatible if and only if for every prime ideal B of R, the ring R/\B is a Dedekind domain. Finally, we show that if R is a UFD such that the free R-module R⊕ R is a semi-compatible module, then R is a Bezout domain.     

特征模对环的刻划

设Ｒ是一个环，Ｍ是一个左Ｒ摸，Ｍ＊＝ＨｏｍＺ（Ｍ，Ｑ／Ｚ）为Ｍ的特征模．Ｒ．Ｒ．Ｃｏｌｂｙ和Ｔ．Ｊ．Ｃｈｏａｔｈａｎ等人利用特征模对ＩＦ环、凝聚环、Ｎｏｅｔｈｅｒ环、Ａｒｔｉｎ环作出了一些非常好的刻划．本文利用特征模对更为广泛的一些环作出了较有意义的刻划．     

Localization of injective modules over valuation rings

It is proved that is injective if is an injective module over a valuation ring , for each prime ideal . Moreover, if or is flat, then is injective, too. It follows that localizations of injective modules over h-local Prüfer domains are injective, too.

     

A generalization of silting modules and Tor-tilting modules

We introduce the concept of weak silting modules, which is a generalization of both silting modules and Tor-tilting modules. It is shown that W is a weak silting module if and only if its character module W⁺ is cosilting. Some properties of weak silting modules are given.     

The local cohomology modules of Matlis reflexive modules are almost cofinite

We show that if and are Matlis reflexive modules over a complete Gorenstein local domain and is an ideal of such that the dimension of is one, then the modules are Matlis reflexive for all and if . It follows that the Bass numbers of are finite. If is not a domain, then the same results hold for .

     

Composition series of modules over Prüfer domains

A weakened version of the Jordan-Hölder theorem is shown to hold for torsion-free finite rank modules over an integral domain precisely when is a Prüfer domain.

     

On flatness properties of cyclic S-posets

In this paper, we discuss flatness properties of cyclic S-posets. As applications, some partially ordered monoids are characterized and some results on S-acts can be also obtained. Research supported by the National Natural Science Foundation of China (No.10626012).     

On co-isosimple modules and co-isoradical of modules

Abstract

In this paper we study “co-isosimple” modules, that is, those modules which are isomorphic to all of its non-zero quotients modules. This allows us to define and study “isomaximal” submodules, “isomax” modules and the “co-isoradical” of a module. We study some of its basic properties and we give a characterization of left V-ring using these concepts.

Communicated by Alberto Facchini     

Tilting modules over path algebras of Dynkin type and complete slice modules

Let T be a tilting A-module over a path algebra of Dynkin type.We prove thatif the indecomposable direct summands of T are in the different (?)-orbits of the AR-quiverof A,then T is a complete slice module.     

Another Gorenstein analogue of flat modules

A right R-module M is called glat if any homomorphism from any finitely presented right R-module to M factors through a finitely presented Gorenstein projective right R-module. The concept of glat modules may be viewed as another Gorenstein analogue of flat modules. We first prove that the class of glat right R-modules is closed under direct sums, direct limits, pure quotients and pure submodules for arbitrary ring R. Then we obtain that a right R-module M is glat if and only if M is a direct limit of finitely presented Gorenstein projective right R-modules. In addition, we explore the relationships between glat modules and Gorenstein flat (Gorenstein projective) modules. Finally we investigate the existence of preenvelopes and precovers by glat and finitely presented Gorenstein projective modules.     

Torsion-free,Divisible, and Mittag-Leffler Modules

We study (relative) 𝒦-Mittag–Leffler modules, with emphasis on the class 𝒦 of absolutely pure modules. A final goal is to describe the 𝒦-Mittag–Leffler abelian groups as those that are, modulo their torsion part, ?₁-free. Several more general results of independent interest are derived on the way. In particular, every flat 𝒦-Mittag–Leffler module (for 𝒦 as before) is Mittag–Leffler. A question about the definable subcategories generated by the divisible modules and the torsion-free modules, resp., has been left open.     

On dual of square free modules

A module M is said to be square free if whenever its submodule is isomorphic to N² = N⊕N for some module N, then N = 0. Dually, a module M is said to be d-square free (dual square free) if whenever its factor module is isomorphic to N² for some module N, then N = 0. In this paper, we give some fundamental properties of d-square free modules and study rings whose d-square free modules are closed under submodules or essential extensions.     

收敛幂级数环的一个性质

本文证明了收敛幂级数环的张量积不是Ｎｏｅｔｈｅｒ环，并提出一个问题。     

平坦的多项式剩余类环

本文证明了如果多项式的剩余类环 A=R[T]/fR[T]作为 R-模是平坦模,且R是约化环,则f是正规多项式.特别地,若R还是连通的,则f的首项系数是单位.也证明了弱整体有限的凝聚环是约化环,以及弱整体为有限的凝聚连通环是整环.     

Projective and generating modules over the ring of pseudorational numbers

The projective, flat, and generating modules over the ring of pseudorational numbers are described. For the projective modules, a complete independent system of invariants is constructed.     

不可分解模的参量化

本文的主要结果是给出了有限群的不可分解模的一种参量化, 它相对于Puig 和Thévenaz 给出的参量化更加自然. 同时我们说明了用这两种不同的方式给出的参量化是一致的.