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 J 2 = 0, where J is the Jacobson radical of R. Finally, we show that every Rad-supplemented module is Rad-⊕-supplemented over dedekind domains.

On a variation of ⊕-supplemented modules

“…( (1) Let R be a local ring which is not field. Combining Theorem 15,[16,Proposition 11] and [13,Corollary 3.3], we have M is ⊕ ss -supplemented if and only if M is isomorphic to a bounded R-module with semisimple radical. (2) Let R be a non-local ring.…”

“…(2) Let R be a non-local ring. By Theorem 15,[13,Theorem 3.2], [12,Proposition 7.3] and [11,Theorem 3.1], M is ⊕ ss -supplemented if and only if M is a torsion module with semisimple radical and every p-component of M is supplemented.…”

“…It follows from [12,Theorem 6.1] that, for a ring R, R P (R) is left perfect, where P (R) is the sum of all left ideals I of R such that I = Rad(I) if and only if every left R-module is Rad-supplemented. In [13], a module M is called Rad-⊕-supplemented if every submodule of M has a Rad-supplement that is a direct summand of M . It is clear that every ⊕-supplemented module is Rad-⊕-supplemented.…”

“…It is clear that every ⊕-supplemented module is Rad-⊕-supplemented. For the concept of Rad-⊕-supplemented, we refer to [14] and [13].…”

I-Rad-⊕-supplemented modules

“…Recall from [6] that a module M is called Rad-⊕-supplemented ( or generalized ⊕-supplemented in [5]) if for every N ⊆ M, there exists a direct summand K of M such that M = N + K and N ∩ K ⊆ Rad(K). In [15], various properties of Rad-⊕-supplemented modules are given. In addition, a ring R is semiperfect if and only if every finitely generated free R-module is generalized ⊕-supplemented (see [5]).…”