Closed Submodules
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.
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SUB MOD U L E S F.Loonstra
The neat subgroups of any abelian group G,introduced by Honda
[6} (see also
Rangaswamy [13,14]and Schoeman C15j),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 Hare not,in general,isomorphic.(ii)The inter-section 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~ ;0, such that N is maximal with respect to this property,and it is denoted by N = pc. A double complement pcc of P is a complement of pC,such that pccJ. P.The starting point is the fact that a submodule N of M is a complement iff N is an(essentially) closed submodul.e 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 sub-modules of a locally uniform module play an important role.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 sub-module of B will be denoted by A feB. §1.General properties. We mention the following results .hl and .1.,.2( see e.g.Goodearl [5 J):
.hl Theorem.If N is a submodule of M,then the following conditions are equivalent: (i) N is a complement for some submodule P~ M;(ii) N is a closed submodule of M; (iii) li P is a complement of N in M,then N is a complement of P:ill M;(iv):i,X N!O LfeM, then ~
L/li
~ e MIN.
Theorem. Let M f e M' and N a submodule of M. Then N is a closed submodule of M iff
N is the intersection of M with a closed submodule N' of M'. A
Since the closed submodules of the injective hull M of M are the injective submodules of M, we find: UaCor.A submodule N Qf M is a closed submodule of M if and only if N is the inter/'-
-section of M with an injective submodule of M.
1...l Definition.A nonzero submodule N of an R-module M is a uniform submodule of Miff any two nonzero submodules of N have nonzero intersection,i.e.iff every nonzero sub-module of N is essential submodule of N. ~
Definition.The R-module M is called a locally uniform
R-module if every nonzero
submodule of M contains a uniform submodule.
l...2 Theorem. The following properties of a submodule
R. Göbel et al. (eds.), Abelian Group Theory © Springer-Verlag Berlin Heidelberg 1983
o,m ~ M are
equivalent:
631
(i)N ~s a minimal closed submodule of M;(ii)N is a closed and unifo
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