Projective ideals of skew polynomial rings over HNP rings

Let R be an hereditary Noetherian prime ring (or, HNP-ring, for short), and let S?=?R[x;σ] be a skew polynomial ring over R with σ being an automorphism on R. The aim of the paper is to describe completely the structure of right projective ideals of R[x;σ] where R is an HNP-ring and to obtain that any right projective ideal of S is of the form X𝔟[x;σ], where X is an invertible ideal of S and 𝔟 is a σ-invariant eventually idempotent ideal of R.     

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．     

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。     

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.     

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,     

Anisotropic Modules over Artinian Principal Ideal Rings

Let V be a finite-dimensional vector space over a field k, and let W be a 1-dimensional k-vector space. Let ?,?: V × V → W be a symmetric bilinear form. Then ?,? is called anisotropic if for all nonzero v ∈ V we have ? v, v ? ≠ 0. Motivated by a problem in algebraic number theory, we give a generalization of the concept of anisotropy to symmetric bilinear forms on finitely generated modules over artinian principal ideal rings. We will give many equivalent definitions of this concept of anisotropy. One of the definitions shows that a form is anisotropic if and only if certain forms on vector spaces are anisotropic. We will also discuss the concept of quasi-anisotropy of a symmetric bilinear form, which has no vector space analogue. Finally, we will discuss the radical root of a symmetric bilinear form, which does not have a vector space analogue either. All three concepts have applications in algebraic number theory.     

投射自由环的Morita特征

本文研究投射自由环的Ｍｏｒｉｔａ特征,证明了投射自由环具有Ｍｏｒｉｔａ不变性．     

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.     

On Projective Modules with Constant Ranks

On Projective Modules with Constant RanksIn this paper,we investigate module structures of rings over which every finitely generated projective module with constant rank is stably free. As applications,we give characterizations of some related rings.     

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.     

Feebly Baer Rings and Modules

A right module M over a ring R is called feebly Baer if, whenever xa = 0 with x ∈ M and a ∈ R, there exists e² = e ∈ R such that xe = 0 and ea = a. The ring R is called feebly Baer if R_R is a feebly Baer module. These notions are motivated by the commutative analog discussed in a recent paper by Knox, Levy, McGovern, and Shapiro [6 Knox , M. L. , Levy , R. , McGovern , W. Wm. , Shapiro , J. ( 2009 ) Generalizations of complemented rings with applications to rings of functions. . J. Alg. Appl. 8 ( 1 ): 17 – 40 .[Crossref] , [Google Scholar]]. Basic properties of feebly Baer rings and modules are proved, and their connections with von Neumann regular rings are addressed.     

Homomorphisms of Projective Remoteness Planes

A characterization of a class of homomorphisms of projective remoteness planes in terms of their coordinate rings is given. A remoteness preserving homomorphism of projective remoteness planes is factored into three homomorphisms of known types. Two of these are constructed from groups associated with the planes and the homomorphism that induces on the coordinate rings of the planes. The third is a covering of planes coordinatized by the same ring. This generalizes known results for projective planes, projective ring planes, and Moufang–Veldkamp planes.     

置换QB-环上的有限生成投射模

研究了置换QB-环上的有限生成投射模,证明了QB-环可以通过有限投射生成子的外替换和内替换来刻画.同时,还研究了QB-环上投射模的消去性.     

On Gorenstein Projective Dimensions of Unbounded Complexes

Let R → S be a ring homomorphism and X be a complex of R-modules. Then the complex of S-modules S?_R^L X in the derived category D(S) is constructed in the natural way. This paper is devoted to dealing with the relationships of the Gorenstein projective dimension of an R-complex X(possibly unbounded) with those of the S-complex S?_R^L X.It is shown that if R is a Noetherian ring of finite Krull dimension and φ : R → S is a faithfully flat ring homomorphism...     

Further Results on Finitely Generated Projective Modules

In this paper, the exchange rings R whose primitive factor rings are artinian are studied. The following results are proved: for any exchange ring R and any two-sided ideal I of R, K ₀(π) : K ₀(R)→K ₀(R/I) is a group epimorphism with the kernel {[P]−[Q] |P = PI, Q = QI}; there is an isomorphism of ordered groups from K ₀(R) to the gorup of all such functions ƒ _P : X→Q(P∈p(R)), where X is the set of all primitive ideals of R and Q, the rational integers. Received February 2, 1999, Accepted December 9, 1999     

关于有限生成投射模的进一步结果

本文研究所有右本原同态象均为阿丁环的调换环,刻画了其K0-群并证明了在有限生成投射模的范畴中关于直和的在同构意义下的n-th root总是唯一的.