No doubt, a notion of the Rad-supplemented modules can constitute a very important situation in the 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 = QUOTE , M is Rad-supplemented 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 QUOTE 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.
Key words: Rad-supplement, Rad-supplemented module, semiperfect ring, artinian ring.
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