Multiplicatively Periodic Rings

Multiplicatively Periodic Rings

On multiplicatively dependent linear numeration systems, and periodic points

Two linear numeration systems, with characteristic polynomial equal to the minimal polynomial of two Pisot numbers β and γ respectively, such that β and γ are multiplicatively dependent, are considered. It is shown that the conversion between one system and the other one is computable by a finite automaton. We also define a sequence of integers which is equal to the number of periodic points of...

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.

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...