ON WEAKLY PERIODIC-LIKE RINGS AND COMMUTATIVITY THEOREMS

A ring R is called periodic if, for every x in R, there exist distinct positive integers m and n such that x(m) = x(n). An element x of R is called potent if x(k) = x for some integer k > 1. A ring R is called weakly periodic if every x in R can be written in the form x = a + b for some nilpotent element a and some potent element b in R. A ring R is called weakly periodic-like if every element x in R which is not in the center C of R can be written in the form x = a + b, with a nilpotent and b potent. Some structure and commutativity theorems are established for weakly periodic-like rings R satisfying certain torsion-freeness hypotheses along with conditions involving some elements being central.