ON w-LOCAL MODULES AND Rad-SUPPLEMENTED MODULES
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All modules considered in this note are over associative com- mutative rings with an identity element. We show that a w-local mod- ule M is Rad-supplemented if and only if M/P(M) is a local module, where P(M) is the sum of all radical submodules of M. We prove that w-local nonsmall submodules of a cyclic Rad-supplemented module are again Rad-supplemented. It is shown that commutative Noetherian rings over which every w-local Rad-supplemented module is supplemented are Artinian. We also prove that if a finitely generated Rad-supplemented module is cyclic or multiplication, then it is amply Rad-supplemented. We conclude the paper with a characterization of finitely generated am- ply Rad-supplemented left modules over any ring (not necessarily com- mutative).
Abstract
All modules considered in this note are over associative com- mutative rings with an identity element. We show that a w-local mod- ule M is Rad-supplemented if and only if M/P(M) is a local module, where P(M) is the sum of all radical submodules of M. We prove that w-local nonsmall submodules of a cyclic Rad-supplemented module are again Rad-supplemented. It is shown that commutative Noetherian rings over which every w-local Rad-supplemented module is supplemented are Artinian. We also prove that if a finitely generated Rad-supplemented module is cyclic or multiplication, then it is amply Rad-supplemented. We conclude the paper with a characterization of finitely generated am- ply Rad-supplemented left modules over any ring (not necessarily com- mutative).
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