Rings With The Symmetric Property for Idempotent-Products
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Let R be a ring with the unity 1, and let e be an idempo- tent of R. In this paper, we discuss some symmetric property for the set {(a1, a2, ..., an) ∈ Rn : a1a2 · · · an = e}. We here investigate some properties of those rings with such a symmetric property for an arbitrary idempotent e; some of our results turn out to generalize some known re- sults observed already when n = 2 and e = 0, 1 by several authors. We also focus especially on the case when n = 3 and e = 1. As consequences of our observation, we also give some equivalent conditions to the commutativity for some classes of rings, in terms of the symmetric property.
Abstract
Let R be a ring with the unity 1, and let e be an idempo- tent of R. In this paper, we discuss some symmetric property for the set {(a1, a2, ..., an) ∈ Rn : a1a2 · · · an = e}. We here investigate some properties of those rings with such a symmetric property for an arbitrary idempotent e; some of our results turn out to generalize some known re- sults observed already when n = 2 and e = 0, 1 by several authors. We also focus especially on the case when n = 3 and e = 1. As consequences of our observation, we also give some equivalent conditions to the commutativity for some classes of rings, in terms of the symmetric property.
참고문헌 (4)
- D. D. Anderson, "Semigroups and rings whose zero products commute", Comm. Algebra, 27, pp.2847~2852 (1999)[DOI]
- P.M. Cohn, "Reversible rings", Bull. London Math. Soc., 31, pp.641~648 (1999)[DOI]
- J. Krempa, "Rings in which annihilators are ideals and their application to semigroup rings", Bull. Acad. Polon. Sci., Math. Astronom. Phys., 25, pp.851~856 (1977)
- T. Y. Lam, "A First Course in Noncommutative Rings" (1991), Springer-Verlag