Düzenli ve kodüzenli alt gruplar

We survey the properties of neat subgroups of abelian groups. Coneat subgroups are always neat subgroups. Conversely, if a torsion group A with all but finitely many primary components zero is a neat subgroup of a group B, then it is coneat in B. We cannot generalize this to any torsion group A. We give another proof for the structure of neat-injective abelian groups using the structure of reduced algebraically compact groups. This proof is generalized to give the structure of c-injective modules over Dedekind domains. A submodule V of a module M is a coneat submodule of M if and only if V is a Rad-supplement of a submodule U of M in M. We investigate some properties of Rad-supplemented modules over any ring R. A module M is Radsupplemented if and only if M/P(M) is Rad-supplemented, where P(M) denotes the sum of all radical submodules of M. A reduced Rad-supplemented module is weakly supplemented. A reduced module is totally Rad-supplemented if and only if it is totally supplemented. A reduced Rad-supplemented module is coatomic. For a left reduced ring R, R is left perfect if and only if every R-module is Rad-supplemented. For a commutative noetherian ring R, if an R-module M is reduced and Rad-supplemented, then it is supplemented. We also investigate some properties of Rad-supplemented modules over discrete valuation rings and Dedekind domains. For a discrete valuation ring R which is not a field, an R-module M is Rad-supplemented if and only if M is the direct sum of its divisible part, a finitely generated free R-module and a bounded R-module. A module M over a Dedekind domain R is Rad-supplemented if and only if the quotient module of M by its divisible part is supplemented. For a Dedekind domain R, a reduced R-module is Rad-supplemented if and only if it is supplemented. Keywords: Rad-supplement, coneat, neat, supplement, complement , c-injective, coatomic module, proper class, closed submodule, high subgroup.