• Title, Summary, Keyword: ascending chain condition on annihilators

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  • Han, Juncheol;Sim, Hyo-Seob
    • East Asian mathematical journal
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    • v.36 no.3
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    • pp.337-347
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    • 2020
  • Let R be a ring with unity, and let I be an ideal of R. Then R is called I-semiregular if for every a ∈ R there exists b ∈ R such that ab is an idempotent of R and a - aba ∈ I. In this paper, basic properties of I-semiregularity are investigated, and some equivalent conditions to the primitivity of e are observed for an idempotent e of an I-semiregular ring R such that I∩eR = (0). For an abelian regular ring R with the ascending chain condition on annihilators of idempotents of R, it is shown that R is isomorphic to a direct product of a finite number of division rings, as a consequence of the observations.