Closed Submodules

Abstract

The neat subgroups of any abelian group G,introduced by Honda [6] (see also Rangaswamy [13,14]and Schoeman [15]),form the collection of all(essentially)closed subgroups of G and they manifest many of the properties of closed submodules of an R-module.E.g.:(i)every subgroup H of a group G possesses a maximal essential extension (=neat hull)of H in G;two neat hulls of H are not,in general,isomorphic.(ii)The intersection of two or more neat subgroups of a group G is not,in general,neat.The closed submodules of an R-module M are related to the complemented submodules N of M.N is a complemented submodule(=complement)of M,if there exists a submodule P of M,P⋂N =0, such that N is maximal with respect to this property,and it is denoted by N = P^c. A double complement P^cc of P is a complement of P^c,such that P^cc⊒ P.The starting point is the fact that a submodule N of M is a complement iff N is an(essentially) closed submodule of M.First of all we pay attention to a characterization of the closed submodules of an R-module,to evt.minimal(resp.maximal)closed submodules and to the intersection property of closed submodules.In §2 the closed submodules are applied in connection with the notion of uniform and locally uniform modules. Attention is paid to the decomposition of locally uniform modules as irrredundant subdirect products of uniform modules. In this connection the maximal closed submodules of a locally uniform module play an important rôle.In §3 we give some results of closed submodules in quasi-injective modules;in §4 we conclude with two applications. All R-modules are unitary left R-modules.The property that A is an essential submodule of B will be denoted by A ⊑_e B.