⊕-Cofinitely Supplemented Modules

Let R be a ring and M a right R-module. M is called -cofinitely supplemented if every submodule N of M with M/N finitely generated has a supplement that is a direct summand of M. In this paper various properties of the -cofinitely supplemented modules are given. It is shown that (1) Arbitrary direct sum of -cofinitely supplemented modules is -cofinitely supplemented. (2) A ring R is semiperfect if and only if every free R-module is -cofinitely supplemented. In addition, if M has the summand sum property, then M is -cofinitely supplemented iff every maximal submodule has a supplement that is a direct summand of M.     

On Graded Distributive Modules

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On strongly ⊕-supplemented modules

Strongly ⊕-supplemented and strongly cofinitely ⊕-supplemented modules are defined and some properties of strongly ⊕-supplemented and strongly cofinitely ⊕-supplemented modules are investigated. Let R be a ring. Then every R-module is strongly ⊕-supplemented if and only if R is perfect. The finite direct sum of ⊕-supplemented modules is ⊕-supplemented. However, this is not true for strongly ⊕-supplemented modules. Any direct sum of cofinitely ⊕-supplemented modules is cofinitely ⊕-supplemented but this is not true for strongly cofinitely ⊕-supplemented modules. We also prove that a supplemented module is strongly ⊕-supplemented if and only if every supplement submodule lies above a direct summand.     

On τ-Extending Modules

Motivated by [2] and [6], we introduce a generalization of extending (CS) modules by using the concept of τ-large submodule which was defined in [9]. We give some properties of this class of modules and study their relationship with the familiar concepts of τ-closed, τ-complement submodules and the other generalization of extending modules (τ-complemented, τ-CS, s−τ-CS modules). We are also interested in determining when a τ-divisible module is τ-extending. For a τ-extending module M with C3, we obtain a decomposition theorem that there is a submodule K of M such that M = t(M) ? KM = \tau (M)\,\oplus\,K and K is τ (M)-injective. We also treat when a direct sum of τ-extending modules is τ-extending.     

On ⌖-supplemented Modules

Let R be a ring and M a right R-module. M is called -supplemented if every submodule of M has a supplement that is a direct summand of M, and M is called completely -supplemented if every direct summand of M is -supplemented. In this paper various properties of these modules are developed. It is shown that (1) Any finite direct sum of -supplemented modules is -supplemented. (2) If M is -supplemented and (D3) then M is completely -supplemented.     

On Flat and Coflat Modules

It is well known that a ring R is left hereditary iff every left ideal of R is projective, iff every submodule of a projective module is projective (ld R≤1), iff every quotient module of an injective module is injective (1cd R≤1), where 1d R and 1cd R means the left global dimension resp. codimension of the ring R. These rings may be generalized to those of weak left global dimension at most 1 (wld R≤1). The latter condition holds iff every left ideal of R is flat, iff every submodule     

On ℳ p -Supplemented Subgroups of Finite Groups

A subgroup H of G is called ? _p -supplemented in G if there exists a subgroup B of G such that G = HB and TB < G for every maximal subgroup T of H with |H: T| =p ^α. In this paper, we investigate the influence of ? _p -supplemented subgroup and some conditions for p-nilpotency and p-supersolvability of finite groups are obtained.     

On Π-Supplemented Subgroups of a Finite Group

A subgroup H of a finite group G is said to satisfy Π-property in G if for every chief factor L/K of G, |G/K: N_G/K(HK/K ∩ L/K)| is a π(HK/K ∩ L/K)-number. A subgroup H of G is called Π-supplemented in G if there exists a subgroup T of G such that G = HT and H ∩ T ≤ I ≤ H, where I satisfies Π-property in G. In this article, we investigate the structure of a finite group G under the assumption that some primary subgroups of G are Π-supplemented in G. The main result we proved improves a large number of earlier results.     

On generalization of ⊕-cofinitely supplemented modules

We study the properties of ⊕-cofinitely radical supplemented modules, or, briefly, cgs ^⊕-modules. It is shown that a module with summand sum property (SSP) is cgs ^⊕ if and only if M/w Loc^⊕ M (w Loc^⊕ M is the sum of all w-local direct summands of a module M) does not contain any maximal submodule, that every cofinite direct summand of a UC-extending cgs ^⊕-module is cgs ^⊕, and that, for any ring R, every free R-module is cgs ^⊕ if and only if R is semiperfect.     

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.     

On Coflat Modules and Their Hereditary Properties

In[4]Hiremath introduced the notion of copureness in the category of R-modules.We define on this basis coflatness of modules and study some of theirhereditary properties dual to that of the flat modules.The later can be used tocharacterize a class of rings larger than that of semihereditary ones.     

A Finite Structure Theorem Between Rings of Endomorphisms of Completely Reducible Modules

In this note we establish a finite structure theorem between rings endomorphisms of homogen-eous completely reducible modules,which generalizes the finite structure theorem between primitiverings as we established before.     

On the Koszul Modules of Exterior Algebras

We prove that the Koszul modules over an exterior algebra can be filtered by the cyclic Koszul modules. We also introduce the cyclic dimension vector as invariants for studying the Koszul modules over an exterior algebra.     

On Multigraded Generalizations of Kirillov–Reshetikhin Modules

We study the category of $\mathbb Z^\ell$ -graded modules with finite-dimensional graded pieces for certain $\mathbb Z^\ell_+$ -graded Lie algebras. We also consider certain Serre subcategories with finitely many isomorphism classes of simple objects. We construct projective resolutions for the simple modules in these categories and compute the Ext groups between simple modules. We show that the projective covers of the simple modules in these Serre subcategories can be regarded as multigraded generalizations of Kirillov–Reshetikhin modules and give a recursive formula for computing their graded characters.     

On Indecomposable Separated Secondary Modules Over ℤ G

In this paper we classify the indecomposable separated secondary modules over G, where G is a cyclic group of prime order and establish a connection between the secondary separated modules and the pure-injective separated modules over such rings.AMS Mathematics Subject Classification (2000) 13F05 16G60     

On the Construction of d—Continuous Modules

In this paper we introduce: and study the notion of dual Gontinuous (d-Contmuous) modnles, A decomposrtion thcorem for a d-continuous modules proved, this generahzes all known decompositron theorems for quasi-projective modules.