Bulletin of the
Korean Mathematical Society BKMS

Characterizations of closed subgroups in abelian groups have been generalized to modules in essentially different ways; they are in general inequivalent. Here we consider the relations between these generalizations over commutative rings, and we characterize the commutative rings over which they coincide. These are exactly the commutative noetherian distributive rings. We also give a characterization of $c$-injective modules over commutative noetherian distributive rings. For a noetherian distributive ring $R$, we prove that, (1) direct product of simple $R$-modules is $c$-injective; (2) an $R$-module $D$ is $c$-injective if and only if it is isomorphic to a direct summand of a direct product of simple $R$-modules and injective $R$-modules.

Keywords: (co)neat submodules, closed submodules, $c$-injective modules, noetherian ring, distributive ring

MSC numbers: 13C13, 13F05, 16D1O