Seçil Çeken

Second Modules Over Noncommutative Rings

Let R be an arbitrary ring. A nonzero unital right R-module M is called a second module if M and all its nonzero homomorphic images have the same annihilator in R. It is proved that if R is a ring such that R/P is a left bounded left Goldie ring for every prime ideal P of R, then a right R-module M is a second module if and only if Q = ann R (M) is a prime ideal of R and M is a divisible right (R/Q)-module. If a ring R satisfies the ascending chain condition on two-sided ideals, then every nonzero R-module has a nonzero homomorphic image which is a second module. Every nonzero Artinian module contains second submodules and there are only a finite number of maximal members in the collection of second submodules. If R is a ring and M is a nonzero right R-module such that M contains a proper submodule N with M/N a second module and M has finite hollow dimension n, for some positive integer n, then there exist a positive integer k ≤ n and prime ideals P i (1 ≤ i ≤ k) such that if L is a proper submodule of M with M/L a second module, then M/L has annihilator P i for some 1 ≤ i ≤ k. Every second submodule of an Artinian module is a finite sum of hollow second submodules.

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.

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.