Characterizing Abelian Generalized Regular Rings that are Noetherian
1
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
A ring R is called generalized regular if for every nonzero x in R there exists y in R such that xy is a nonzero idempotent. In this paper, we observe some equivalent conditions for the generalized regular rings that are abelian in terms of idempotents, and we also investigate the primitivity of an idempotent for such a ring. By using the investigation, we characterize such a kind of rings that are noetherian by showing that an abelian generalized regular ring R is noetherian if and only if R is isomorphic to a direct product of finitely many division rings. We also observe some interesting consequences of our results.
Abstract
A ring R is called generalized regular if for every nonzero x in R there exists y in R such that xy is a nonzero idempotent. In this paper, we observe some equivalent conditions for the generalized regular rings that are abelian in terms of idempotents, and we also investigate the primitivity of an idempotent for such a ring. By using the investigation, we characterize such a kind of rings that are noetherian by showing that an abelian generalized regular ring R is noetherian if and only if R is isomorphic to a direct product of finitely many division rings. We also observe some interesting consequences of our results.
참고문헌 (6)
- D. D. Anderson, "Semigroups and rings whose zero products commute", Comm. in Algebra, 27, pp.2847~2852 (1999)
- 한준철, "On idempotents in relation with regularity", 대한수학회지, 53(1), pp.217~232 (2016), 대한수학회
- 한준철, "Rings With The Symmetric Property for Idempotent-Products", East Asian Mathematical Journal, 34(5), pp.615~621 (2018), 영남수학회[DOI]
- C. Huh, "Basic examples and extensions of symmetric rings", J. Pure and Applied Algebra, 202, pp.154~167 (2005)
- 정다운, "On properties related to reversible rings", 대한수학회보, 52(1), pp.247~261 (2015), 대한수학회
- T. Y. Lam, "A First Course in Noncommutative Rings" (1991), Springer-Verlag