On Weak Dual Rickart Modules and Dual Baer Modules

We introduce and study the notion of wd-Rickart modules (i.e. modules M such that for every nonzero endomorphism ? of M, the image of ? contains a nonzero direct summand of M). We show that the class of rings R for which every right R-module is wd-Rickart is exactly that of right semi-artinian right V-rings. We prove that a module M is dual Baer if and only if M is wd-Rickart and M has the strong summand sum property. Several structure results for some classes of wd-Rickart modules and dual Baer modules are provided. Some relevant counterexamples are indicated.     

On Clean Rings

A ring R is called clean if every element of R is the sum of an idempotent and a unit. Let M be a R-module. It is obtained in this article that the endomorphism ring End(M) is clean if and only if, whenever A = M′ ⊕ B = A₁ ⊕ A₂ with M′ ? M, there is a decomposition M′ =M₁ ⊕ M₂ such that A = M′ ⊕ [A₁ ∩ (M₁ ⊕ B)] ⊕ [A₂ ∩ (M₂ ⊕ B)]. Then unit-regular endomorphism rings are also described by direct decompositions.     

有限置换模自同态环的可分性质

设Q是有限置换右R模,则End_R(Q)是可分环当且仅当对所有A,B∈FP(Q),A AA B B B A≤ B或 B≤A,作为应用得到了 End_R(P Q)是可分环当且仅当End_R P和End_R Q为可分环,其中P,Q为有限置换右R模。     

Endomorphism Rings

In this article we construct an affine prime PI ring with a cyclic module whose endomorphism ring is not Jacobson. Then, using similar techniques, we construct an affine prime Noetherian PI ring with a cyclic module whose endomorphism ring is not Noetherian.     

A-Solvability and Mixed Abelian Groups

An abelian group A is an S-group (S ⁺-group) if every subgroup B ≤ A of finite index is A-generated (A-solvable). This article discusses some of the differences between torsion-free S-groups and mixed S-groups, and studies (mixed) S- and S ⁺-groups, which are self-small and have finite torsion-free rank.     

A Note on Finitely A-Presented Abelian Groups

The class of mixed groups  was introduced by Glaz and Wickless. This article investigates subclasses  of  such that the functor Hom(A, ?)/t Hom(A, ?) between  and the category of right E(A)/tE(A)-modules is full.     

ON THE FINITENESS PROPERTIES OF THE GENERALIZED LOCAL COHOMOLOGY MODULES

Abstract

The question, posed in Crawley and Jónsson (Crawley, P., Jónsson, B. (1964). Refinements for infinite direct decompositions of algebraic systems. Pacific J. Math. 14:797–855), whether the finite exchange property implies the unrestricted exchange property for any modules, is still open. In this note we obtain the equivalence of the finite exchange property and the unrestricted exchange property for the class of modules whose endomorphism rings are Abelian.     

On a class of coherent regular rings

The paper investigates a special class of quasi-local rings. It is shown that these rings are coherent and regular in the sense that every finitely generated submodule of a free module has a finite free resolution.

     

GENERALIZED POWER SERIES MODULES

In this paper we obtain results pertaining to noetherian nature of generalised power series modules over rings not necessarily possessing an indentity element. These considerably strengthen earlier results of Ribenboim on this topic.     

𝔏-Rickart Modules

It is well known that the Rickart property of rings is not a left-right symmetric property. We extend the notion of the left Rickart property of rings to a general module theoretic setting and define 𝔏-Rickart modules. We study this notion for a right R-module M _R where R is any ring and obtain its basic properties. While it is known that the endomorphism ring of a Rickart module is a right Rickart ring, we show that the endomorphism ring of an 𝔏-Rickart module is not a left Rickart ring in general. If M _R is a finitely generated 𝔏-Rickart module, we prove that End _R (M) is a left Rickart ring. We prove that an 𝔏-Rickart module with no set of infinitely many nonzero orthogonal idempotents in its endomorphism ring is a Baer module. 𝔏-Rickart modules are shown to satisfy a certain kind of nonsingularity which we term “endo-nonsingularity.” Among other results, we prove that M is endo-nonsingular and End _R (M) is a left extending ring iff M is a Baer module and End _R (M) is left cononsingular.     

交换环上三角矩阵模间的线性保持映射(英文)

本文研究了交换环上三角矩阵模间的线性保持问题.利用矩阵计算技巧和局部化技巧,刻画了上三角矩阵T_n(R)上分别保持立方幂等,{1}逆,{1,2}逆和群逆的所有R模自同构集合中的元素,其中R是交换环.     

A new setting for constructing von Neumann regular rings

A classical and clever construction by George Bergman in the mid 1970s was of a unit-regular algebra Q (over an arbitrary field F) that contains a regular subalgebra R which is not unit-regular. The algebra Q was constructed inside a full linear ring of an uncountable-dimensional vector space over F. Here we give a much simpler construction of Bergman’s R and Q inside an algebra B of countably-infinite matrices. The algebra B does not seem to have been noticed before, possibly because it is not obviously a ring. It also suggests possibilities for answering the difficult open problem of constructing a non-separative regular ring.     

A SURJECTIVE HOMOMORPHISM OF LOCAL COHOMOLOGY MODULES

Let R be a local ring and M a finitely generated generalized Cohen-Macaulay R-module such that dim _R M = dim _R M/αM + height_Mα a for all ideals α of R. Suppose that H_I ^j(M) ≠ 0 for an ideal I of R and an integer j > height_M I. We show that there exists an ideal J ? such that a. height_M J = j;

b. the natural homomorphism H_I ^j(M) → H_I ^j(M) is an isomorphism, for all i > j; and,

c. the natural homomorphism H_I ^j(M) → H_I ^j(M) is surjective.

By using this theorem, we obtain some results about Betti numbers, coassociated primes, and support of local cohomology modules.     

弱$r$-Clean环

As generalization of r-clean rings and weakly clean rings, we define a ring R is weakly r-clean if for any a∈R there exist an idempotent e and a regular element r such that a = r + e or a = r-e. Some properties and examples of weakly r-clean rings are given. Furthermore, we prove the weakly clean rings and weakly r-clean rings are equivalent for abelian rings.     

$GP$-$V^{'}$-环的非奇异性与正则性

In this paper,we introduce a non-trivial generalization of ZI-rings-quasi ZI-rings.A ring R is called a quasi ZI-ring,if for any non-zero elements a,b ∈ R,ab = 0 implies that there exists a positive integer n such that an = 0 and anRbn = 0.The non-singularity and regularity of quasi ZI,GP-Vˊ-rings are studied.Some new characterizations of strong regular rings are obtained.These effectively extend some known results.     

Countable linear transformations are clean

It is shown that every linear transformation on a vector space of countable dimension is the sum of a unit and an idempotent.