On torsion-free periodic rings

There is a great deal of literature on periodic rings, respectively, torsion-free rings (especially of rank two). The aim of this paper is to provide a link between these two topics. All groups considered here are Abelian, with addition as the group operation. By order of an element we always mean the additive order of this element. All rings are associative but not necessarily with identity. The additive group of the ring R will be denoted by R+. n(R) denotes the ring of all the n×n matrices with entries in R. A ring R is called periodic if for each x ∈ R, the set {x,x2,x3, . . .} is finite, or equivalently, for each x ∈ R there are positive integers m(x), n(x) such that xm(x) = xm(x)+n(x). However, periodic rings can also be defined (see [20]) by requiring that (i) the multiplicative semigroup of R is periodic, or, (ii) if a∈ R, then a power of a generates a finite subring. Examples of periodic rings are finite rings, nil rings, and direct sums of matrix rings over finite fields. Z, the ring of all the integers, is not periodic. Research on periodic rings (the term “periodic” seems to have been first used by Chacron [16]) was mainly done in two directions: (i) finding sufficient conditions on periodic rings which imply commutativity, Bell being the prominent name in this direction (all over the last 40 years; e.g., see [10, 11, 12]) but also Abu-Khuzam and Yaqub (see [1, 2, 13, 26]), respectively, (ii) finding structure results for some special classes of periodic rings (e.g., see [3, 5, 12]). However, it should be noticed that the starting point for these investigations was the Jacobson theorem, whose proof contains many ideas which could be used also in more general contexts. For later convenience we state here some elementary properties for a periodic ring. (iii) Any infinite-order element is a zero divisor (in the subring generated by itself). (iv) Every idempotent in R has finite order. (v) For each a∈ R some power of a is idempotent.