Ergül Türkmen

On some new operations in soft module theory

In this paper, we introduce the notions of sum and direct sum of soft submodules, small soft submodules and radical of a soft module. Moreover, we obtain basic properties of such soft submodules.

Some properties of Rad-supplemented modules

module theory, because of a generalization of the notion of supplemented modules. Therefore, our work presents a key role mainly in some properties and characterizations of Rad-supplement submodules and Rad-supplemented modules. In this paper, we show that, for a duo module M = , M is Radsupplemented if and only if each Mi is Rad-supplemented. Moreover, we prove that if an R-module M contains an artinian submodule N, M is Rad-supplemented if and only if is Rad-supplemented. In addition, a left hereditary ring R is Rad-supplemented if and only if it is semiperfect, and if the ring is commutative, R is artinian if and only if every left R-module is (amply) Rad-supplemented. We also provide various properties of semilocal modules.

RAD- ⊕ -SUPPLEMENTED MODULES

Abstract In this paper we provide various properties of Rad-⊕-supplemented modules. In particular, we prove that a projective module M is Rad- ⊕-supplemented if and only if M is ⊕-supplemented, and then we show that a commutative ring R is an artinian serial ring if and only if every left R-module is Rad-⊕-supplemented. Moreover, every left R-module has the property (P*) if and only if R is an artinian serial ring and J2 = 0, where J is the Jacobson radical of R. Finally, we show that every Rad-supplemented module is Rad-⊕-supplemented over dedekind domains.

Modules that have a supplement in every cofinite extension

Abstract. Let R be a ring and M a left R-module. An R-module N is called a cofinite extension of M in case and is finitely generated. We say that M has the property CE (resp. CEE) if M has a supplement (resp. ample supplements) in every cofinite extension. In this study we give various properties of modules with these properties. We show that a module M has the property CEE iff every submodule of M has the property CE. A ring R is semiperfect iff every left R-module has the property CE. We also study cofinitely injective modules, direct summands of every cofinite extension, as a generalization of injective modules.