Communicated by Efim Zelmanov
Keywords:Injective module Poor module Injectivity domain V-, QI-, SI-, PCI-, QF-ring In a recent paper, Alahmadi, Alkan and López-Permouth defined a module M to be poor if M is injective relative only to semisimple modules, and a ring to have no right middle class if every right module is poor or injective. We prove that every ring has a poor module, and characterize rings with semisimple poor modules. Next, a ring with no right middle class is proved to be the ring direct sum of a semisimple Artinian ring and a ring T which is either zero or of one of the following types: (i) Morita equivalent to a right PCI-domain, (ii) an indecomposable right SI-ring which is either right Artinian or a right V-ring, and such that soc (T T ) is homogeneous and essential in T T and T has a unique simple singular right module, or (iii) an indecomposable right Artinian ring with homogeneous right socle coinciding with the Jacobson radical and the right singular ideal, and with unique non-injective simple right module. In case (iii) either T T is poor or T is a QF-ring with J (T ) 2 = 0. Converses of these cases are discussed. It is shown, in particular, that a QF-ring R with J (R) 2 = 0 and homogeneous right socle has no middle class.
. We show that the β * relation is an equivalence relation and has good behaviour with respect to addition of submodules, homomorphisms and supplements. We apply these results to introduce the class of G * -supplemented modules and to investigate this class and the class of H-supplemented modules. These classes are located among various well-known classes of modules related to the class of lifting modules. Moreover these classes are used to explore an open question of S. H. Mohamed and B. J. Mueller. Examples are provided to illustrate and delimit the theory.2000 Mathematics Subject Classification. 16D10, 16D50.
Abstract. In this work, we say submodules X and Y of M are β * g equivalence, Xβ * g Y , if and onlyIt is proved that the β * g relation is an equivalent relation and has good behaviour with respect to addition of submodules and homomorphisms.
for every submodule X of M there is a supplement submodule S of M such that Xβ * S. In this work some new properties of β * are given and G * -supplemented modules are studied. Also completely G * -supplemented modules and G * -radical supplemented modules are defined.
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