A commutativity condition for s -unital rings

LetR be ans-unital ring, and we prove a commutativity theorem ofR satisfying the following conditions: (1) For eachx, y εR, there exist bounded positive integersk=k(x,y), s=s(x,y), t=t(x,y) (where, at least one ofk, s, t is not equal to 1) such that (xy)k=x ^sy^t, (xy)k+1=x ^s+1 y ^t+1; (2)N, the set of all nilpotent elements ofR, isp-torsion free, wherep is the L. C. M. (least common multiple) of allk, s, t. Project supported by The Science Foundation of NECC for Returns Synopsis of the first author Yin Zhiyun, professor, born in May, 1960, study fields are commutative algebras and economical mathematics.