I-Rad-⊕-supplemented modules

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Let M be an R-module and I be an ideal of R. We say that M is I-Rad-⊕-supplemented, provided for every submodule N of M, there exists a direct summand K of M such that M = N + K, N ∩ K ⊆ IK and N ∩ K Rad(K). The aim of this paper is to show new properties of I-Rad-⊕-supplemented modules. Especially, we show that any finite direct sum of I-Rad-⊕-supplemented modules is I-Rad-⊕-supplemented. We also prove that an R-module M is I-Rad-⊕-supplemented if and only if K and MK are I-Rad-⊕-supplemented for a fully invariant direct summand K of M. Finally, we determine the structure of I-Rad-⊕-supplemented modules over a discrete valuation ring.

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