Mustafa Alkan

Semiperfect Modules with Respect to a Preradical

ABSTRACT In this article, we consider the module theoretic version of I-semiperfect rings R for an ideal I which are defined by Yousif and Zhou (2002). Let M be a left module over a ring R, N ∈ σ[M], and τ M a preradical on σ[M]. We call N τ M -semiperfect in σ[M] if for any submodule K of N, there exists a decomposition K = A ⊕ B such that A is a projective summand of N in σ[M] and B ≤ τ M (N). We investigate conditions equivalent to being a τ M -semiperfect module, focusing on certain preradicals such as Z M , Soc, and δ M . Results are applied to characterize Noetherian QF-modules (with Rad(M) ≤ Soc(M)) and semisimple modules. Among others, we prove that if every R-module M is Soc-semiperfect, then R is a Harada and a co-Harada ring.

Semiregular Modules with Respect to a Fully Invariant Submodule

Abstract Let M be a left R-module and F a submodule of M for any ring R. We call M F-semiregular if for every x ∈ M, there exists a decomposition M = A ⊕ B such that A is projective, A ≤ Rx and Rx ∩ B ≤ F. This definition extends several notions in the literature. We investigate some equivalent conditions to F-semiregular modules and consider some certain fully invariant submodules such as Z(M), Soc(M), δ(M). We prove, among others, that if M is a finitely generated projective module, then M is quasi-injective if and only if M is Z(M)-semiregular and M ⊕ M is CS. If M is projective Soc(M)-semiregular module, then M is semiregular. We also characterize QF-rings R with J(R)2 = 0.

On the second spectrum and the second classical Zariski topology of a module

Let R be an associative ring with identity and Specs(M) denote the set of all second submodules of a right R-module M. In this paper, we investigate some interrelations between algebraic properties of a module M and topological properties of the second classical Zariski topology on Specs(M). We prove that a right R-module M has only a finite number of maximal second submodules if and only if Specs(M) is a finite union of irreducible closed subsets. We obtain some interrelations between compactness of the second classical Zariski topology of a module M and finiteness of the set of minimal submodules of M. We give a connection between connectedness of Specs(M) and decomposition of M for a right R-module M. We give several characterizations of a noetherian module M over a ring R such that every right primitive factor of R is artinian for which Specs(M) is connected.

Dual of Zariski Topology for Modules

In this paper we introduce the dual Zariski topology on the set of second submodules of M, denoted by Specs(M), for an R‐module M. We give some relationships between Specs(M) and Spec(R/Ann(M)). By using this topological space, we give some characterizations of rings and modules.

Generating function for q-Eulerian polynomials and their decomposition and applications

The aim of this paper is to define a generating function for q-Eulerian polynomials and numbers attached to any character χ of the finite cyclic group G. We derive many functional equations, q-difference equations and partial deferential equations related to these generating functions. By using these equations, we find many properties of q-Eulerian polynomials and numbers. Using the generating element of the finite cyclic group G and the generating element of the subgroups of G, we show that the generating function with a conductor f can be written as a sum of the generating function with conductors which are less than f.MSC:05A40, 11B83, 11B68, 11S80.

On Radical Formula over Free Modules with Two Generators

In this paper, we study on the module M = R⊕R over a commutative ring R with identity. After characterizing the radical of a submodule N of M, we give some conditions for N to satisfy the radical formula. In particular, we show that R⊕R satisfy the radical formula if R is an arithmetical ring.

On strongly 2-absorbing second submodules

In this paper, we study on the concept of strongly 2-absorbing second submodule which is a dual notion of 2-absorbing submodule and a generalization of second submodule. We give some properties and characterizations of this submodule class and investigate the relationships with second and secondary submodules.In this paper, we study on the concept of strongly 2-absorbing second submodule which is a dual notion of 2-absorbing submodule and a generalization of second submodule. We give some properties and characterizations of this submodule class and investigate the relationships with second and secondary submodules.

The structure of some topologies on a module

In this paper, we construct some topologies on a module M by using the dual Zariski topology on a quotient module of M and define a continuous map between these topologies. Then we find a topology on M which is homeomorphic to the dual Zariski topology on a quotient module of M.

On submodule characterization and decomposition of modules over group rings

In this paper, for a commutative unity ring R and a finite group G, we characterize some submodule properties of RG– module MG introduced in [8]. Our aim is to decompose MG into RG–submodules by defining an idempotent in EndRG MG and to verify the relation between the quotient of group module and the quotient of related module.

Singular and nonsingular modules relative to a torsion theory

ABSTRACT In this paper, we introduce and study torsion-theoretic generalizations of singular and nonsingular modules by using the concept of τ-essential submodule for a hereditary torsion theory τ. We introduce two new module classes called τ-singular and non-τ-singular modules. We investigate some properties of these module classes and present some examples to show that these new module classes are different from singular and nonsingular modules. We give a characterization of τ-semisimple rings via non-τ-singular modules. We prove that if M∕τ(M) is non-τ-singular for a module M, then every submodule of M has a unique τ-closure. We give some properties of the torsion theory generated by the class of all τ-singular modules. We obtain a decomposition theorem for a strongly τ-extending module by using non-τ-singular modules.