Journal of the
Korean Mathematical Society JKMS

All modules considered in this note are over associative commutative rings with an identity element. We show that a $w$-local module $M$ 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 amply $Rad$-supplemented left modules over any ring (not necessarily commutative).

Keywords: $w$-local modules, $Rad$-supplemented modules, amply $Rad$-supplemented modules

MSC numbers: 16D10, 16D80