Rafail Alizade

MODULES WHOSE MAXIMAL SUBMODULES HAVE SUPPLEMENTS

A ring R is semiperfect if and only if, for every (cyclic)R-module M, every maximal submodule has (ample) supplements in M. A non-local commutative domain R is h-local if and only if, for every (cyclic) torsion R-module M, every maximal submodule of M has (ample) supplements in M.

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.

Special precovers in cotorsion theories

A cotorsion theory is defined as a pair of classes Ext-orthogonal to each other. We give a hereditary condition (HC) which is satisfied by the (flat, cotorsion) cotorsion theory and give properties satisfied by arbitrary cotorsion theories with an HC. Given a cotorsion theory with an HC, we consider the class of all modules having a special precover with respect to the first class in the cotorsion theory and show that this class is closed under extensions. We then raise the question of whether this class is resolving or coresolving. AMS 2000 Mathematics subject classification: Primary 18G15. Secondary 18E40

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.

Modules and abelian groups with minimal (pure-) projectivity domains

In this paper, we give a complete description of the projectively poor abelian groups and prove that there exists a pure projectively poor abelian group. We show that over a commutative Artinian ring every module having a projectively poor factor module by a pure submodule, is itself projectively poor. We also give some other properties of pure projectively poor modules.

The Proper Class Generated by Weak Supplements

We show that, for hereditary rings, the smallest proper classes containing respectively the classes of short exact sequences determined by small submodules, submodules that have supplements and weak supplement submodules coincide. Moreover, we show that this class can be obtained as a natural extension of the class determined by small submodules. We also study injective, projective, coinjective and coprojective objects of this class. We prove that it is coinjectively generated and its global dimension is at most 1. Finally, we describe this class for Dedekind domains in terms of supplement submodules.

Abelian groups whose nonzero endomorphisms have nonessential kernels

In this paper, we describe completely the 𝒦-singular subgroup of an abelian group and a 𝒦-nonsingular abelian group in terms of the basic subgroups of its p-components and the quotient group by the torsion part. We also prove that a pure subgroup and a quotient group by a pure subgroup of a 𝒦-nonsingular abelian group are 𝒦-nonsingular and give a condition under which a pure extension of a 𝒦-nonsingular abelian group by a 𝒦-nonsingular group is 𝒦-nonsingular.

RESOLUTIONS IN COTORSION THEORIES

We consider the λ− (μ‐) and λ‐ (μ‐) dimensions of modules taken under a cotorsion theory (F, C) satisfying the Hereditary Condition, and establish some inequalities between the dimensions of the modules of a short exact sequence, not necessarily Hom (F, −) exact. We investigate the question of whether the property of having a (special) F‐ or C‐resolution of length n is resolving, closed under extensions or coresolving and establish some inequalities connecting the λ‐ (μ‐) and λ‐ (μ‐) dimensions of modules in a short exact sequence.