The Cotorsionfree Radical of Relatively Torsion Theory

Letτbe a torsion theory on R-mod and M be a left R-mod. In this paper theτ-cotorsionfree radical Cτ(M) of M is studied. When r is stable, the construction and the supplementing radical of Cτ(M) are obtained.     

$f$-投射与$f$-内射模

Let R be a ring. A fight R-module M is called f-projective if Ext^1 （M, N） = 0 for any f-injective right R-module N. We prove that （F-proj,F-inj） is a complete cotorsion theory, where （F-proj （F-inj） denotes the class of all f-projective （f-injective） right R-modules. Semihereditary rings, von Neumann regular rings and coherent rings are characterized in terms of f-projective modules and f-injective modules.     

相对n-FP-内射模

给出了n-FP-内射模的定义,M为左R-模,如果对任意的左R-模N有Ext1(N,M)=0,则称M为n-FP-内射模,作为应用,给出了n-FP-内射模的一些等价条件.     

拟内射模中的消去定理

当左拟内射模Ｍ的自同态环Ｅｎｄ_RＭ为一Ｄｅｃｋｉｎｄ有限环时．Ｍ的任何两个相互同构的子模的左相关补子横也同构。     

倾斜RnA-模及其导出的挠理论

本文通过函子Ｔ＝－ＡＲｎＡ讨论了倾斜Ａ 模与倾斜ＲｎＡ 模的重要联系，推广了［１］的主要结果；讨论了倾斜ＲｎＡ 模ＴＸ与倾斜Ａ 模导出的挠理论在相同性和分裂性等方面的关系．     

伪i-内射半模

主要引进了伪i-内射半模的定义,并根据对偶原则,参照k-投射半模及内射模的结论,得到了伪i-内射半模的一些很好的性质,从而实现了把环中内射模的某屿性质在半环中内射半模方面的部分推广.     

关于拟GP-内射模

在本文中,我们定义了拟GP-内射模,并且得到了关于它的一些结果.这些结果总结了GP-内射环和拟P-内射模的一些结果.     

对偶扩张代数的分裂挠理论与Generic模

设A是一个有限维代数,R是A的对偶扩张代数.MA是一个A-模.给定一个倾斜R-模M(○)AR,我们知道MA一定是一个倾斜A-模.设(TM(○)AR,FM(○)AR)与(TM,FM)是分别由M (○)AR和MR导出的挠理论.本文讨论挠理论的分裂性以及Generic A-模与Generic R-模之间的关系。     

对偶扩张代数的分裂挠理论与Generic模

设A是一个有限维代数,R是A的对偶扩张代数.MA是一个A-模.给定一个倾斜R-模M（?）AR,我们知道MA一定是一个倾斜A-模 设（TM（?）AR,FM（?）AR）与（TM,FM）是分别由M（?）AR和MR导出的挠理论.本文讨论挠理论的分裂性以及GenericA-模与GenericR-模之间的关系.     

内射模的t－平坦性

设Ｒ为环，ｔ是左Ｒ－模范畴的一个遗传挠理论．文中证明了下述各点等价：（１）每个内射左Ｒ－模是ｔ－平坦的；（２）每个ｔ－有限表现左Ｒ－模的内射包络是ｔ－平坦的；（３）每个ｔ－有限表现左Ｒ－模是自由Ｒ－模的子模；（４）每个ｔ－有限表现左Ｒ－模是自反的且其对偶模是Ｈ－有限生成的．     

The Torsion Theory Generated by <Emphasis Type="Italic">M</Emphasis>-Small Modules

Let M be a right R-module, the class of all M-small modules, and P a projective cover of M in [M]. We consider the torsion theories = ( ), = ( ), and = ( ) in [M], where is the torsion theory generated by is the torsion theory cogenerated by , and is the dual Lambek torsion theory. We study some conditions for to be cohereditary, stable, or split, and prove that Rej(M, ) = M = (= = ) = Gen_M(P) .2000 Mathematics Subject Classification: 16S90     

Tensor and Torsion Products of Relative Injective Modules with Respect to a Semidualizing Module

Let R be a commutative Cohen–Macaulay ring, and let C be a semidualizing module of R. In this paper, we show that C is generically dualizing if and only if the tensor products of injective and C-injective R-modules are injective. This leads to a characterization of dualizing modules as well as generalizes a result of Enochs and Jenda.     

Extending modules relative to a torsion theory

An R-module M is said to be an extending module if every closed submodule of M is a direct summand. In this paper we introduce and investigate the concept of a type 2 τ-extending module, where τ is a hereditary torsion theory on Mod-R. An R-module M is called type 2 τ-extending if every type 2 τ-closed submodule of M is a direct summand of M. If τ _I is the torsion theory on Mod-R corresponding to an idempotent ideal I of R and M is a type 2 τ _I -extending R-module, then the question of whether or not M/MI is an extending R/I-module is investigated. In particular, for the Goldie torsion theory τ _G we give an example of a module that is type 2 τ _G -extending but not extending.     

Semiperfect Modules with Respect to a Preradical

In an earlier paper [8] the authors introduced strongly and properly semiprime modules. Here properly semiprime modules M are investigated under the condition that every cyclic submodule is M-projective (self-pp-modules). We study the idempotent closure of M using the techniques of Pierce stalks related to the central idempotents of the self-injective hull of M. As an application of our theory we obtain several results on (not necessarily associative) biregular, properly semiprime, reduced and Firings. An example is given of an associative semiprime PSP ring with polynomial identity which coincides with its central closure and is not biregular (see 3.6). Another example shows that a semiprime left and right FP-injective Pl-ring need not be regular (see 4.8). Some of the results were already announced in [7].     

Relative Projective Modules and Relative Injective Modules

Let R be a ring, and n and d fixed non-negative integers. An R-module M is called (n, d)-injective if Ext ^d+1 _R (P, M) = 0 for any n-presented R-module P. M is said to be (n, d)-projective if Ext¹ _R (M, N) = 0 for any (n, d)-injective R-module N. We use these concepts to characterize n-coherent rings and (n, d)-rings. Some known results are extended.     

Idempotent and Nilpotent Submodules of Multiplication Modules

All rings are commutative with identity, and all modules are unital. The purpose of this article is to investigate multiplication von Neumann regular modules. For this reason we introduce the concept of nilpotent submodules generalizing nilpotent ideals and then prove that a faithful multiplication module is von Neumann regular if and only if it has no nonzero nilpotent elements and its Krull dimension is zero. We also give a new characterization for the radical of a submodule of a multiplication module and show in particular that the radical of any submodule of a Noetherian multiplication module is a finite intersection of prime submodules.