# RINGS WITH THE SYMMETRIC PROPERTY FOR IDEMPOTENT-PRODUCTS

• Han, Juncheol (Department of Mathematics Education, Pusan National University) ;
• Sim, Hyo-Seob (Department of Applied Mathematics, Pukyong National University)
• Accepted : 2018.08.31
• Published : 2018.09.30

#### Abstract

Let R be a ring with the unity 1, and let e be an idempotent of R. In this paper, we discuss some symmetric property for the set $\{(a_1,a_2,{\cdots},a_n){\in}R^n:a_1a_2{\cdots}a_n=e\}$. 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 results 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.

#### Acknowledgement

Supported by : Pukyong National University

#### References

1. D. D. Anderson and V. Camillo, Semigroups and rings whose zero products commute, Comm. Algebra 27 (1999), 2847-2852. https://doi.org/10.1080/00927879908826596
2. P.M. Cohn, Reversible rings, Bull. London Math. Soc. 31 (1999), 641-648. https://doi.org/10.1112/S0024609399006116
3. J. Krempa and D. Niewieczerzal, Rings in which annihilators are ideals and their application to semigroup rings, Bull. Acad. Polon. Sci., Math. Astronom. Phys. 25 (1977), 851-856.
4. T. Y. Lam, A First Course in Noncommutative Rings, Springer-Verlag, New York, 1991.