On Copure Projective Modules and Copure Projective Dimensions

Let R be any ring. A right R-module M is called n-copure projective if Ext¹(M, N) = 0 for any right R-module N with fd(N) ≤ n, and M is said to be strongly copure projective if Ext ⁱ (M, F) = 0 for all flat right R-modules F and all i ≥ 1. In this article, firstly, we present some general properties of n-copure projective modules and strongly copure projective modules. Then we define and investigate copure projective dimensions of modules and rings. Finally, more properties and applications of n-copure projective modules, strongly copure projective modules and copure projective dimensions are given over coherent rings with finite self-FP-injective dimension.     

Projective Modules and Complete Intersections

Let A be a Noetherian ring of dimension n and P be a projective A module of rank n having trivial determinant. It is proved that if n is even and the image of a generic element g P^* is a complete intersection, then [P] = [Q A] in K₀(A) for some projective A module Q of rank n – 1. Further, it is proved that if n is odd, A is Cohen–Macaulay and [P] = [Q A] in K₀(A) for some projective A module Q of rank n – 1, then P has a unimodular element.     

Forcing Linearity Numbers for Multiplication Modules

In this article, we prove that for any multiplication module M, the forcing linearity number of M, fln(M), belongs to {0,1,2}, and if M is finitely generated whose annihilator is contained in only finitely many maximal ideals, then fln(M) = 0. Also, the forcing linearity numbers of multiplication modules over some special rings are given. We also show that every multiplication module is semi-endomorphal.     

关于R—投射模是投射模的环

本文研究了所有Ｒ—投射模都是投射模的环（ＲＰ—环），得出了它的几个等价条件，证明了：Ｓ＝Ｒｎ为ＲＰ—环当且仅当Ｒ为ＲＰ—环；∑ｎｉ＝１Ｒｉ为ＲＰ—环当且仅当每个Ｒｉ为ＲＰ—环．讨论了ＲＰ—环的左投射维数．     

7c-Gorenstein Projective, ;Cc-Gorenstein Injectiveand Wcr-Gorenstei丨丨 Flat Modules

The authors introduce and investigate the Tc-Gorenstein projective, Lc- Gorenstein injective and Hc-Gorenstein flat modules with respect to a semidualizing module C which shares the common properties with the Gorenstein projective, injective and flat modules, respectively. The authors prove that the classes of all the Tc-Gorenstein projective or the Hc-Gorenstein flat modules are exactly those Gorenstein projective or flat modules which are in the Auslander class with respect to C, respectively, and the classes of all the Lc-Gorenstein ＇injective modules are exactly those Gorenstein injective modules which are in the Bass class, so the authors get the relations between the Gorenstein projective, injective or flat modules and the C-Gorenstein projective, injective or flat modules. Moreover, the authors consider the Tc（R）-projective and Lc（R）-injective dimensions and Tc（R）-precovers and Lc（R）-preenvelopes. Fiually, the authors study the Hc-Gorenstein flat modules and extend the Foxby equivalences.     

有常秩的投射模

本文定义更具一般性的模(未必是有限生成投射模)的常秩的概念,并证明了如果M有常秩n,∧~n M是有限生成的,则M是有限生成的,还证明了若M是有常秩n的投射模,则M一定是有限生成的。     

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.     

Some Remarks on Generalized GCD Domains

An integral domain R is a generalized GCD (GGCD) domain if the semigroup of invertible ideals of R is closed under intersection. In this article we extend the definition of PF-prime ideals to GGCD domains and develop a theory of these ideals which allows us to characterize Prüfer and π -domains among GGCD domains. We also introduce the concept of generalized GCD modules as a natural generalization of GGCD domains to the module case. An R-module M is a GGCD module if the set of invertible submodules of M is closed under intersection. We show that an integral domain R is a GGCD domain if and only if a faithful multiplication R-module M is a GGCD module. Various properties and characterizations of faithful multiplication GGCD modules over integral domains are considered and consequently, necessary and sufficient conditions for a ring R(M), the idealization of M, to be a GGCD ring are given.     

局部几乎完全整环的几点注记

整环$R$被称为局部几乎完全整环, 指的是对任意极大理想${\frak m}$都有$R_{{\frak m}}$是几乎完全整环. 本文利用局部完全环, 几乎投射模, 弱内射模, 几乎强平坦模和强Matlis余挠模给出了局部几乎完全整环的若干等价刻画.     

Gorenstein Modules and Dualizing Modules

In this article, we study the characterizations of Gorenstein injective left S-modules and finitely generated Gorenstein projective left R-modules when there is a dualizing S-R-bimodule associated with a right noetherian ring R and a left noetherian ring S.     

Generalization of Regular Modules

We study generalizations of regular modules by Ramamurthy and Mabuchi. These are also generalizations of fully right idempotent and fully left idempotent rings, respectively. We also define and study the properties of *-weakly regular modules, a generalization of fully idempotent rings.     

Frobenius扩张上的$n$-Gorenstein投射模和维数

In this paper, we study n-Gorenstein projective modules over Frobenius extensions and n-Gorenstein projective dimensions over separable Frobenius extensions. Let R ■ A be a Frobenius extension of rings and M any left A-module. It is proved that M is an n-Gorenstein projective left A-module if and only if A ■_RM and Hom_R(A, M) are n-Gorenstein projective left A-modules if and only if M is an n-Gorenstein projective left R-module. Furthermore, when R ■ A is a separable Frobenius extension, n-Gorenstein projective dimensions are considered.     

投射Schur模与合成因子

本文用不可分解投射Ｓｃｈｕｒ模研究并刻划了有限型自入射代数的不可分解模由合成因子决定的条件。     

Multiplication Modules in Which Every Prime Submodule is Contained in a Unique Maximal Submodule

Abstract

Let R be a commutative ring. An R-module M is called a multiplication module if for each submodule N of M, N?=?IM for some ideal I of R. An R-module M is called a pm-module, i.e., M is pm, if every prime submodule of M is contained in a unique maximal submodule of M. In this paper the following results are obtained. (1) If R is pm, then any multiplication R-module M is pm. (2) If M is finitely generated, then M is a multiplication module if and only if Spec(M) is a spectral space if and only if Spec(M)?=?{PM?|?P?∈?Spec(R) and P???M ^⊥}. (3) If M is a finitely generated multiplication R-module, then: (i) M is pm if and only if Max(M) is a retract of Spec(M) if and only if Spec(M) is normal if and only if M is a weakly Gelfand module; (ii) M is a Gelfand module if and only if Mod(M) is normal. (4) If M is a multiplication R-module, then Spec(M) is normal if and only if Mod(M) is weakly normal.     

分次直投射模及应用

本文引进分次直投射模的概念,得到分次直投射模的一个判定定理,并利用分次直投射模刻划了分次左遗传环,分次左半遗传环,分次左半单环和分次左PP-环,     

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.     

Some Aspects of Strongly P-projective Modules and Homological Dimensions

A left R-module M is called strongly P-projective if Extⁱ(M, P) = 0 for all projective left R-modules P and all i ≥ 1. In this article, we first discuss properties of strongly P-projective modules. Then we introduce and study the strongly P-projective dimensions of modules and rings. The relations between the strongly P-projective dimension and other homological dimensions are also investigated.     

关于G－Matlis自反模的几点注记

本文证明，如果Ｒ是一个Ｎｏｅｔｈｅｒ完备半局部环，则Ｒ－模Ｍ是Ｎｏｅｔｈｅｒ（Ａｒｔｉｎ）模当且仅当对任意ＡｒｔｉｎＲ－模Ｎ，Hom _R(M, N)是Ａｒｔｉｎ（Ｎｏｅｔｈｅｒ）模．     

Noether环的几点注记

本文利用特征模给出了Ｎｏｅｔｈｅｒ环的一些充分必要条件，并由此讨论了Ｎｏｅｔｈｅｒ环上的内射模与平坦模的一些性质