Rings whose modules have maximal or minimal injectivity domains

JOURNAL OF ALGEBRA

卷:330

Article

Er, Noyan¹  Lopez-Permouth, Sergio²  Sokmez, Nurhan³ 

[1] Univ Rio Grande, Dept Math, Rio Grande, OH 45674 USA

[2] Ohio Univ, Dept Math, Athens, OH 45701 USA

[3] Ondokuz Mayis Univ, Dept Math, Samsun, Turkey

关键词: Injective module;    Poor module;    Injectivity domain;    V-, QI-, SI-, PCI-, QF-ring;   

DOI  :  10.1016/j.jalgebra.2010.10.038

来源: Elsevier

【 摘 要 】

In a recent paper, Alahmadi. Alkan and Lopez-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. (C) 2010 Elsevier Inc. All rights reserved.

【 授权许可】

Free