On Purity and Quasiequality of Abelian Groups

We study the Cohn purity in an abelian group regarded as a left module over its endomorphism ring. We prove that if a finite rank torsion-free abelian group G is quasiequal to a direct sum in which all summands are purely simple modules over their endomorphism rings then the module _E(G) G is purely semisimple. This theorem makes it possible to construct abelian groups of any finite rank which are purely semisimple over their endomorphism rings and it reduces the problem of endopure semisimplicity of abelian groups to the same problem in the class of strongly indecomposable abelian groups.     

The Zero-Divisor Graphs Which Are Uniquely Determined By Neighborhoods

A nonempty simple connected graph G is called a uniquely determined graph, if distinct vertices of G have distinct neighborhoods. We prove that if R is a commutative ring, then Γ(R) is uniquely determined if and only if either R is a Boolean ring or T(R) is a local ring with x² = 0 for any x ∈ Z(R), where T(R) is the total quotient ring of R. We determine all the corresponding rings with characteristic p for any finite complete graph, and in particular, give all the corresponding rings of K_n if n + 1 = p^q for some primes p, q. Finally, we show that a graph G with more than two vertices has a unique corresponding zero-divisor semigroup if G is a zero-divisor graph of some Boolean ring.     

Conjugacy of idempotents in jordan pairs

The injectives in the category of associative rings and homomorphisms with accessible images are investigated.It is shown that every such ring has an identity and is a direct sum of finitely many indecomposable injectives.The indecomposable injectives are shown to be simple when they are algebras over fields other than Z_p and in the remaining characteristic p case to have only three ideals.A necessary and sufficient condition is obtained for certain finite direct sums of indecomposable injectives to be injective.     

The lattice structure of preradicals III: operators

We continue the study of the big lattice of preradicals over a ring. We consider several operators acting on this lattice, for instance the pseudocomplement, the annihilator and the totalizer, as well as some relations among them. Using some of these operators we give characterizations of V-rings, of rings that are a finite direct sum of injective hulls of simple modules, and of rings such that besides the latter condition have also the property that each pair of simple modules are homologically connected.     

On a class of non-noetherian V-ring

Right V-rings R with infinitely generated right socle SOC(R_R) such that R/SOC(R_R) is a division ring are characterized as those non-noetherian rings over which a cyclic right module is either non-singular or injective. Furthermore, it is shown that a non-noetherian, right V-ring S is Morita-equivalent to a ring of this type iff all singular simple right S-modules are isomorphic and every direct sum of uniform modules with an injective module over S is extending.     

Noetherian rings with almost injective simple modules

We characterize right Noetherian rings over which all simple modules are almost injective. It is proved that R is such a ring, if and only if, the complements of semisimple submodules of every R-module M are direct summands of M, if and only if, R is a finite direct sum of right ideals I_r, where I_r is either a Noetherian V-module with zero socle, or a simple module, or an injective module of length 2. A commutative Noetherian ring for which all simple modules are almost injective is precisely a finite direct product of rings R_i, where R_i is either a field or a quasi-Frobenius ring of length 2. We show that for commutative rings whose all simple modules are almost injective, the properties of Kasch, (semi)perfect, semilocal, quasi-Frobenius, Artinian, and Noetherian coincide.     

Group graded rings

Let R be a group graded ring . The map ( , ): R × R →₁ defined by: (x,y) = (xy)₁ , is an inner product on R. In this paper we investigate aspects of nondegeneracy of the product, which is a generalization of the notion of strongly G —graded rings,introduced by Dade. We show that various chain conditions are satisfied by R if and only if they are satisfied by R₁ , and that when R₁ is simple artinian, then R is a crossed product R₁ ^* G. We give conditions for simple R-modules to be completely reducible R₁ -modules . Finally, we prove an incomparability theorem,when G is finite abelian.     

Commutative rings whose proper ideals are direct sums of uniform modules

An interesting result, obtaining by some theorems of Asano, Köthe and Warfield, states that: “for a commutative ring R, every module is a direct sum of uniform modules if and only if R is an Artinian principal ideal ring.” Moreover, it is observed that: “every ideal of a commutative ring R is a direct sum of uniform modules if and only if R is a finite direct product of uniform rings.” These results raise a natural question: “What is the structure of commutative rings whose all proper ideals are direct sums of uniform modules?” The goal of this paper is to answer this question. We prove that for a commutative ring R, every proper ideal is a direct sum of uniform modules, if and only if, R is a finite direct product of uniform rings or R is a local ring with the unique maximal ideal ? of the form ? = U⊕S, where U is a uniform module and S is a semisimple module. Furthermore, we determine the structure of commutative rings R for which every proper ideal is a direct sum of cyclic uniform modules (resp., cocyclic modules). Examples which delineate the structures are provided.     

Links ore sets,and classical localizations

The notion of a simple ring D_Gderived from a group ring KG is introduced in case K is a field and G is an infinite residually finite group. The close link between D_Gand K_G is exploited in both directions: first, for a simple proof of the Kaplansky's conjecture concerning direct finiteness of KG. Second, to show that D_Gprovides counter-examples to some conjectures dealing with von Neumann regular rings and the rings all of whose one-sided ideals are generated by idempotents.     

关于半clean群环

环$R$称为是半clean的, 是指环中的每个元素都是一个单位与一个周期元的和. clean环是半clean的. 刻画半clean群环的一般情形是不容易的. 我们的目的是考虑如下问题：若$G$ 是局部有限群或者是阶是3的循环群, 群环$RG$何时是semiclean的. clean群环上的一些已有结果被推广.     

Constacyclic codes over finite commutative semi-simple rings

Finite commutative semi-simple rings are direct sum of finite fields. In this study, we investigate the algebraic structure of λ-constacyclic codes over such finite semi-simple rings. Among others, necessary and sufficient conditions for the existence of self-dual, LCD, and Hermitian dual-containing λ-constacyclic codes over finite semi-simple rings are provided. Using the CSS and Hermitian constructions, quantum MDS codes over finite semi-simple rings are constructed.     

A note on indecomposable modules

In this note we study rings having only a finite number of non isomorphic uniform modules with non zero socle. It is proved that a commutative ring with this property is a direct sum of a finite ring and a ring of finite representation type. In the non commutative case we show that most P.I. rings having only a finite number of non isomorphic modules with non zero socle are in fact artinian.     

Commutative Local Rings Whose Ideals are Direct Sums of Cyclic Modules

A well-known result of Köthe and Cohen-Kaplansky states that a commutative ring R has the property that every R-module is a direct sum of cyclic modules if and only if R is an Artinian principal ideal ring. This motivated us to study commutative rings for which every ideal is a direct sum of cyclic modules. Recently, in Behboodi et al. Commutative Noetherian local rings whose ideals are direct sums of cyclic modules (J. Algebra 345:257–265, 2011) the authors considered this question in the context of finite direct products of commutative Noetherian local rings. In this paper, we continue their study by dropping the Noetherian condition.     

无穷矩阵环上的导子

讨论无穷矩阵环上的导子,证明了环R上有限个元素不为零的无穷矩阵坏的每个导子均可表示为两个特殊导子之和。     

A new characterisation of hereditary pi rings

In this paper we prove a new characterisation of hereditary PI rings, namely we show that a Noetherian, but not Artinian, PI ringR that is an order in an Artinian ring splits into a direct sum of an Artinian ring of finite representation type and hereditary semiprime rings if and only if all its proper Artinian factor rings are of finite representation type. We also show, through examples, that the above characterisation does not hold for some more general settings. Supported by the EC via TMR-Fellowship ERB4001GT63713.     

Finite rings in which commutativity is transitive

A ring is called commutative transitive if commutativity is a transitive relation on its nonzero elements. Likewise, it is weakly commutative transitive (wCT) if commutativity is a transitive relation on its noncentral elements. The main topic of this paper is to describe the structure of finite wCT rings. It is shown that every such ring is a direct sum of an indecomposable noncommutative wCT ring of prime power order, and a commutative ring. Furthermore, finite indecomposable wCT rings are either two-by-two matrices over fields, local rings, or basic rings with two maximal ideals. We characterize finite local rings as generalized skew polynomial rings over coefficient Galois rings; the associated automorphisms of the Galois ring give rise to a signature of the local ring. These are then used to further describe the structure of finite local and wCT basic rings.     

环的广义Noether性(英)

众所周知,环R是右Noether的当且仅当任意内射右R-模的直和是内射的．本文我们将用Ne-内射模和U-内射模来刻画Ne-Noether环和U-Noether环．     

Decomposing torsion modules

E. Matlis proved that if R is an integral domain with quotient field Q and K is the R-module Q/R, then all torsion R-modules decompose into a direct sum of local submodules if and only if K decomposes into a direct sum of local submodules. Thus K is a test module to determine whether torsion modules decompose. We generalize this result to commutative rings. If R is a commutative ring and a torsion theory of R is given by a Gabriel topology , then form the ring of quotients R_ and let K be the cokernel of the canonical ring homomorphism from R to R_. In some special cases, every -torsion R-module decomposes into a direct sum of local submodules if and only if K decomposes. However, there is an example where this is not the case. The principal result is: given R,  and K, there is a related filter _K of ideals of R, which is a subset of , such that all _K-pretorsion R-modules decompose into a direct sum of local submodules if and only if K decomposes. The relationship between  and _K is investigated.     

STRONGLY PRIME GROUP RINGS

Abstract

A ring R is (right) strongly prime (SP) if every nonzero two sided ideal contains a finite set whose right annihilator is zero, SP rings have been studied by Handelman and Lawrence who raised the problem of characterizing SP group algebras. They showed that if R is SP and G is torsion free Abelian, then the group ring RG is SP. The aim of this note is to determine some more group rings which are SP.

For a ring R we also define the strongly prime radical s(R). We then show that s(R)G = s(W) for certain classes of groups.