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Ergül Türkmen

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.

Open access

Ergül Türkmen

Abstract

It is known that a commutative ring R is an artinian principal ideal ring if and only if every left R-module is ⊕-supplemented. In this paper, we show that a commutative ring R is a semiperfect principal ideal ring if every left R-module is ⊕-cofinitely supplemented. The converse holds if R is a max ring. Moreover, we study maximally ⊕- supplemented modules as a proper generalization of ⊕-cofinitely supplemented modules. Using these modules, we also prove that a ring R is semiperfect if and only if every projective left R-module with small radical is supplemented.

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Burcu Nişanci Türkmen and Ergül Türkmen

Abstract

In this paper, over an arbitrary ring we define the notion of weakly radical supplemented modules (or briefly wrs-module), which is adapted from Zöschinger’s radical supplemented modules over a discrete valuation ring (DVR), and obtain the various properties of these modules. We prove that a wrs-module having a small radical is weakly supplemented. Moreover, we show that a ring R is left perfect if and only if every left R-module is wrs. Also, we prove that every wrs-module over a DVR is radical supplemented.