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.     

Rickart Modules

The concept of right Rickart rings (or right p.p. rings) has been extensively studied in the literature. In this article, we study the notion of Rickart modules in the general module theoretic setting by utilizing the endomorphism ring of a module. We provide several characterizations of Rickart modules and study their properties. It is shown that the class of rings R for which every right R-module is Rickart is precisely that of semisimple artinian rings, while the class of rings R for which every free R-module is Rickart is precisely that of right hereditary rings. Connections between a Rickart module and its endomorphism ring are studied. A characterization of precisely when the endomorphism ring of a Rickart module will be a right Rickart ring is provided. We prove that a Rickart module with no infinite set of nonzero orthogonal idempotents in its endomorphism ring is precisely a Baer module. We show that a finitely generated module over a principal ideal domain (PID) is Rickart exactly if it is either semisimple or torsion-free. Examples which delineate the concepts and results are provided.     

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.     

On 𝒦-Nonsingular Modules and Applications

We introduce the notion of 𝒦-nonsingularity of a module and show that the class of 𝒦-nonsingular modules properly contains the classes of nonsingular modules and of polyform modules. A necessary and sufficient condition is provided to ensure that this property is preserved under direct sums. Connections of 𝒦-nonsingular modules to their endomorphism rings are investigated. Rings for which all modules are 𝒦-nonsingular are precisely determined. Applications include a type theory decomposition for 𝒦-nonsingular extending modules and internal characterizations for 𝒦-nonsingular continuous modules which are of type I, type II, and type III, respectively.     

关于Gorensteincotorsion模

讨论了Gorensteincotorsion模与内射模之间的关系,证明了R是GorensteinvonNeumann正则环当且仅当任意R模M的Oorensteincotorsion包络与内射包络是同构的,当且仅当E（M）／M是Gorenstein平坦模,同时,也讨论了Gorensteincotorsion模与cotorsion模之间的联系。     

Weak Rickart and dual weak Rickart objects in abelian categories

We introduce and investigate weak relative Rickart objects and dual weak relative Rickart objects in abelian categories. Several types of abelian categories are characterized in terms of (dual) weak relative Rickart properties. We relate our theory to the study of relative regular objects and (dual) relative Baer objects. We also give some applications to module and comodule categories.     

Annihilator Ideal-Based Zero-Divisor Graphs Over Multiplication Modules

For a commutative ring R with identity, an ideal-based zero-divisor graph, denoted by Γ _I (R), is the graph whose vertices are {x ∈ R?I | xy ∈ I for some y ∈ R?I}, and two distinct vertices x and y are adjacent if and only if xy ∈ I. In this article, we investigate an annihilator ideal-based zero-divisor graph by replacing the ideal I with the annihilator ideal Ann(M) for a multiplication R-module M. Based on the above-mentioned definition, we examine some properties of an R-module over a von Neumann regular ring, and the cardinality of an R-module associated with Γ _Ann(M)(R).     

关于内射模的P-平坦性

通过引用P-平坦模的定义,引入了右IPF环的概念,推广了右IF环的概念,这对研究IF环及QF环具有重要的作用,同时对右IPF环的性质作了一些刻画,得到了右IPF环的若干个等价命题;最后,用P-平坦模及右IPF环推出了正则环的一些等价条件．     

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.     

Strongly Rickart objects in abelian categories

We introduce and study (dual) strongly relative Rickart objects in abelian categories. We prove general properties, we analyze the behaviour with respect to (co)products, and we study the transfer via functors. We also give applications to Grothendieck categories, (graded) module categories and comodule categories. Our theory of (dual) strongly relative Rickart objects may be employed in order to study strongly relative regular objects and (dual) strongly relative Baer objects in abelian categories.     

MORPHIC MODULES

A module M is called morphic if M/M α ? ker(α) for all endomorphisms α in end(M), and a ring R is called a left morphic ring if _RR is a morphic module. We consider the open question when the matrix ring M_n(R) is left morphic by relating it to when Rⁿ is morphic as a left R-module. More generally, we investigate when M being morphic implies that end(M) is left morphic, and conversely. Finally, we relate the morphic condition to internal cancellation in the module.     

Strongly Rickart objects in abelian categories: Applications to strongly regular and strongly Baer objects

We show how the theory of (dual) strongly relative Rickart objects may be employed in order to study strongly relative regular objects and (dual) strongly relative Baer objects in abelian categories. For each of them, we prove general properties, we analyze the behavior with respect to (co)products, and we study the transfer via functors. We also give applications to Grothendieck categories, (graded) module categories and comodule categories.     

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.     

t-Extending Modules and t-Baer Modules

We introduce the notions of “t-extending modules,” and “t-Baer modules,” which are generalizations of extending modules. The second notion is also a generalization of nonsingular Baer modules. We show that a homomorphic image (hence a direct summand) of a t-extending module and a direct summand of a t-Baer module inherits the property. It is shown that a module M is t-extending if and only if M is t-Baer and t-cononsingular. The rings for which every free right module is t-extending are called right Σ-t-extending. The class of right Σ-t-extending rings properly contains the class of right Σ-extending rings. Among other equivalent conditions for such rings, it is shown that a ring R is right Σ-t-extending, if and only if, every right R-module is t-extending, if and only if, every right R-module is t-Baer, if and only if, every nonsingular right R-module is projective. Moreover, it is proved that for a ring R, every free right R-module is t-Baer if and only if Z ₂(R _R ) is a direct summand of R and every submodule of a direct product of nonsingular projective R-modules is projective.     

Products of projections in von Neumann algebras

We describe the elements of von Neumann algebras which can be represented as products of orthogonal projections and idempotents, and estimate the minimal number of terms in the product.     

关于全不变扩张环和模(英文)

In this paper, we discuss FI-extending property of rings and modules. The main results are the following： a characterization of von Neumann regular rings which are two-sided FI-extending is given; sufficient conditions for direct summands of FI-extending modules to be FI-extending are obtained; and at last, a necessary and sufficient condition for nonsingular modules over nonsingular rings to be FI-extending is given.     

Soc-Injective Rings and Modules*

If M and N are right R-modules, M is called Socle-N-injective (Soc-N-injective) if every R-homomorphism from the socle of N into M extends to N. Equivalently, for every semisimple submodule K of N, any R-homomorphism f : K → M extends to N. In this article, we investigate the notion of soc-injectivity.     

拟-Armendariz模

For a right R-module N, we introduce the quasi-Armendariz modules which are a common generalization of the Armendariz modules and the quasi-Armendariz rings, and investigate their properties. Moreover, we prove that NR is quasi-Armendariz if and only if Mm(N)Mm(R) is quasi-Armendariz if and only if Tm(N)Tm(R) is quasi-Armendariz, where Mm(N) and Tm(N) denote the m×m full matrix and the m×m upper triangular matrix over N, respectively. NR is quasi-Armendariz if and only if N[x]R[x] is quasi-Armendariz. It is shown that every quasi-Baer module is quasi-Armendariz module.     

关于对偶模的平坦性和内射性(Ⅱ)

对交换环Ｒ和Ｂ－模范畴上的一个内射余生成元Ｂ,我们用相对于Ｅ的对偶模的性质刻画了ＱＦ环,ＩＦ环和半遗传环．