Rings with finiteness conditions on certain subsets

Abstract

Finiteness conditions on a ring or on certain subsets of a ring have several implications on the structure of the ring. This study of finiteness conditions was motivated by a well-known theorem of Wedderburn which states that a finite division ring must be a field. In one of our results in this study, we prove that a finite ring which is multiplicatively generated by idempotent elements must be Boolean. We also study the structure of rings having at most finitely many nonnilpotent elements. Indeed, we prove that a ring having at most finitely many nonnilpotent elements must be either nil or finite. We also show that a ring with a finite number of non-central elements is either finite or commutative. Since every finite ring is clearly periodic, we consider a more general class of the above rings, which are rings satisfying the property that for each x∊R , either x is periodic or there exists a positive integer K=K(x) such that x(to the power k) ∊C (where C is the center of R ) for all k≥K. We study the structure of certain classes of these rings. In particular, we show that a prime ring satisfying the above property is either commutative or periodic.