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On (semi)regularity and the total of rings and modules

Authors
Journal
Journal of Algebra
0021-8693
Publisher
Elsevier
Publication Date
Volume
322
Issue
2
Identifiers
DOI: 10.1016/j.jalgebra.2009.03.020
Keywords
  • Semipotent Ring
  • Semiregular Ring
  • The Total
  • Jacobson Radical
  • (Co)Singular Ideal
  • Endomorphism Ring
  • [Formula Omitted]

Abstract

Abstract Let M and N be two modules over a ring R. Recent works by Kasch, Schneider, Beidar, Mader, and others have shown that some of the ring and module theory can be carried out in the context of hom R ( M , N ) . The study of substructures of hom R ( M , N ) such as the radical, the singular and co-singular ideals and the total has raised new questions for research in this area. This paper is a continuation of study of these substructures, focusing on when the total is equal to the radical, as well as their connections with (semi)regularity of hom R ( M , N ) . New results obtained include necessary and sufficient conditions for the total to equal the radical, a description of the maximal regular sub-bimodule of hom R ( M , N ) , the existence of the maximal semiregular ideal of a ring, and answers to a number of existing open questions.

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