Engin Büyükaşık

Cofinitely Weak Supplemented Modules

Abstract We prove that a module M is cofinitely weak supplemented or briefly cws (i.e., every submodule N of M with M/N finitely generated, has a weak supplement) if and only if every maximal submodule has a weak supplement. If M is a cws-module then every M-generated module is a cws-module. Every module is cws if and only if the ring is semilocal. We study also modules, whose finitely generated submodules have weak supplements.

Poor and pi-poor Abelian groups

ABSTRACT In this paper, poor abelian groups are characterized. It is proved that an abelian group is poor if and only if its torsion part contains a direct summand isomorphic to , where P is the set of prime integers. We also prove that pi-poor abelian groups exist. Namely, it is proved that the direct sum of U(ℕ), where U ranges over all nonisomorphic uniform abelian groups, is pi-poor. Moreover, for a pi-poor abelian group M, it is shown that M can not be torsion, and each p-primary component of M is unbounded. Finally, we show that there are pi-poor groups which are not poor, and vise versa.

Absolutely s-Pure Modules and Neat-Flat Modules

Let R be a ring with an identity element. We prove that R is right Kasch if and only if injective hull of every simple right R-modules is neat-flat if and only if every absolutely pure right R-module is neat-flat. A commutative ring R is hereditary and noetherian if and only if every absolutely s-pure R-module is injective and R is nonsingular. If every simple right R-module is finitely presented, then (1) R R is absolutely s-pure if and only if R is right Kasch and (2) R is a right -CS ring if and only if every pure injective neat-flat right R-module is projective if and only if every absolutely s-pure left R-module is injective and R is right perfect. We also study enveloping and covering properties of absolutely s-pure and neat-flat modules. The rings over which every simple module has an injective cover are characterized.

Rugged modules: The opposite of flatness

ABSTRACT Relative notions of flatness are introduced as a mean to gauge the extent of the flatness of any given module. Every module is thus endowed with a flatness domain and, for every ring, the collection of flatness domains of all of its modules is a lattice with respect to class inclusion. This lattice, the flatness profile of the ring, allows us, in particular, to focus on modules which have a smallest flatness domain (namely, one consisting of all regular modules.) We establish that such modules exist over arbitrary rings and we call them Rugged Modules. Rings all of whose (cyclic) modules are rugged are shown to be precisely the von Neumann regular rings. We consider rings without a flatness middle class (i.e., rings for which modules must be either flat or rugged.) We obtain that, over a right Noetherian ring every left module is rugged or flat if and only if every right module is poor or injective if and only if R = S×T, where S is semisimple Artinian and T is either Morita equivalent to a right PCI-domain, or T is right Artinian whose Jacobson radical properly contains no nonzero ideals. Character modules serve to bridge results about flatness and injectivity profiles; in particular, connections between rugged and poor modules are explored. If R is a ring whose regular left modules are semisimple, then a right module M is rugged if and only if its character left module M+ is poor. Rugged Abelian groups are fully characterized and shown to coincide precisely with injectively poor and projectively poor Abelian groups. Also, in order to get a feel for the class of rugged modules over an arbitrary ring, we consider the homological ubiquity of rugged modules in the category of all modules in terms of the feasibility of rugged precovers and covers for arbitrary modules.

Neat-flat Modules

Let R be a ring. A right R-module M is said to be neat-flat if the kernel of any epimorphism Y → M is neat in Y, i.e., the induced map Hom(S, Y) → Hom(S, M) is surjective for any simple right R-module S. Neat-flat right R-modules are projective if and only if R is a right -CS ring. Every cyclic neat-flat right R-module is projective if and only if R is right CS and right C-ring. It is shown that, over a commutative Noetherian ring R, (1) every neat-flat module is flat if and only if every absolutely coneat module is injective if and only if R ≅ A × B, wherein A is a QF-ring and B is hereditary, and (2) every neat-flat module is absolutely coneat if and only if every absolutely coneat module is neat-flat if and only if R ≅ A × B, wherein A is a QF-ring and B is Artinian with J 2(B) = 0.

On w-local modules and Rad-supplemented modules

All modules considered in this note are over associative com- mutative rings with an identity element. We show that a w-local mod- ule 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 am- ply Rad-supplemented left modules over any ring (not necessarily com- mutative). All rings considered in this paper will be commutative with an identity element (except for Section 5) and all modules will be left unitary modules. Unless otherwise stated R denotes an arbitrary commutative ring. Let M be an arbitrary R-module. We will denote by Rad(M) the Jacobson radical of M. A submodule L of M is called small in M (notation L ≪ M) if M 6 L+N for every proper submodule N of M. The annihilator of M in R will be denoted by AnnR(M) = {� ∈ R : �x = 0 for all x ∈ M} and for every element x of M, the annihilator of x is denoted by AnnR(x) = {� ∈ R : �x = 0}. A module M is said to be radical if Rad(M) = M. The sum of all radical submodules of a module M will be denoted by P(M). A module M is said to be reduced if P(M) = 0. We say that the ring R is reduced if the R-module RR is reduced. For submodules U and V of a module M, the submodule V is said to be a Rad- supplement of U in M if U+V = M and U∩V ⊆ Rad(V ). A module M is called Rad-supplemented if every submodule of M has a Rad-supplement in M. On the other hand, a submodule N of M is said to have ample Rad-supplements in

CONEAT SUBMODULES AND CONEAT-FLAT MODULES

Abstract. A submodule N of a right R-module M is called coneat iffor every simple right R-module S, any homomorphism N → S can beextended to a homomorphism M → S. M is called coneat-flat if thekernel of any epimorphism Y →M →0 is coneat in Y. It is proven that(1) coneat submodules of any right R-module are coclosed if and only ifR is right K-ring; (2) every right R-module is coneat-flat if and only ifR is right V-ring; (3) coneat submodules of right injective modules areexactly the modules which have no maximal submodules if and only ifR is right small ring. If R is commutative, then a module M is coneat-flat if and only if M + is m-injective. Every maximal left ideal of R isfinitely generated if and only if every absolutely pure left R-module is m-injective. A commutative ring Ris perfect if and only if every coneat-flatmodule is projective. We also study the rings over which coneat-flat andflat modules coincide. 1. IntroductionA subgroup A of an abelian group B is said to be neat in B if pA = A∩pBfor every prime integer p. The notion of neat subgroup was generalized tomodules by Renault (see, [12]). Namely, a submodule N of a right R-moduleM is called neat in M, if for every simple right R-module S, Hom(S,M) →Hom(S,M/N) → 0 is epic. Dually, in [8], a submodule N of a right R-moduleM is called coneat in M if Hom(M,S) → Hom(N,S) → 0 is epic for everysimple right R-module S. The notions of neat and coneat are coincide overthe ring of integers. By [8, Theorem], the commutative domains over whichneat and coneat submodules coincide are exactly the domains with finitely gen-erated maximal ideals (i.e., N-domains). This result was extended to certaincommutative rings in [5]. Recently, modules related to neat and coneat sub-modules are considered by several authors. In [5], a right R-module M is calledabsolutely neat (resp. coneat) if M is a neat (resp. coneat) submodule of anymodule containing it. According to [16], a right R-module M is m-injective