Simple periodic rings

On Periodic Rings

It is proved that a ring is periodic if and only if, for any elements x and y , there exist positive integers k,l,m, and n with either k =m or l =n, depending on x and y , for which xkyl = xmyn. Necessary and sufficient conditions are established for a ring to be a direct sum of a nil ring and a J-ring. 2000 Mathematics Subject Classification. Primary 16U99, 16N20, 16D70.

Multiplicatively Periodic Rings

On Generalized Periodic-Like Rings

Let R be a ring with center Z, Jacobson radical J , and set N of all nilpotent elements. Call R generalized periodic-like if for all x ∈ R \ (N ∪ J ∪ Z) there exist positive integers m, n of opposite parity for which xm − xn ∈ N ∩ Z. We identify some basic properties of such rings and prove some results on commutativity. Let R be a ring; and let N = N(R), Z = Z(R) and J = J(R) denote respective...

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. T...

On Structure of Certain Periodic Rings and Near-rings

The aim of this work is to study a decomposition theorem for rings satisfying either of the properties xy = xpf(xyx)xq or xy = xpf(yxy)xq , where p = p(x,y), q = q(x,y) are nonnegative integers and f(t)∈ tZ[t] vary with the pair of elements x,y, and further investigate the commutativity of such rings. Other related results are obtained for near-rings.