RINGS WITH THE SYMMETRIC PROPERTY FOR IDEMPOTENT-PRODUCTS

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.