Hazar Abu-Khuzam

Rings and groups with commuting powers

Let n be a fixed positive integer. Let R be a ring with identity which satisfies (i) xnyn=ynxn for all x,y in R, and (ii) for x,y in R, there exists a positive integer k=k(x,y) depending on x and y such that xkyk=ykxkand (n,k)=1. Then R is commutative. This result also holds for a group G. It is further shown that R and G need not be commutative if any of the above conditions is dropped.

Structure of rings with certain conditions on zero divisors

Let R be a ring such that every zero divisor x is expressible as a sum of a nilpotent element and a potent element of R:x=a

A weak periodicity condition for rings.

A ring is called semi-weakly periodic if each element which is not in the center or the Jacobson radical can be written as the sum of a potent element and a nilpotent element. After discussing some basic properties of such rings, we investigate their commutativity behavior.

On rings with prime centers

Let R be a ring, and let C denote the center of R. R is said to have a prime center if whenever ab belongs to C then a belongs to C or b belongs to C. The structure of certain classes of these rings is studied, along with the relation of the notion of prime centers to commutativity. An example of a non-commutative ring with a prime center is given.

A NOTE ON RINGS WHICH ARE MULTIPLICATIVELY GENERATED BY IDEMPOTENTS AND NILPOTENTS

We give the structure of certain rings which are multiplicatively generated by nilpotents or multiplicatively generated by idempotents and nilpotents.

Periodic rings with commuting nilpotents

Let R be a ring (not necessarily with identity) and let N denote the set of nilpotent elements of R. Suppose that (i) N is commutative, (ii) for every x in R, there exists a positive integer k=k(x) and a polynomial f(λ)=fx(λ) with integer coefficients such that xk=xk

Commutativity and structure of rings with commuting nilpotents

Let R be a ring and let N denote the set of nilpotent elements of R. Let Z denote the center of R. Suppose that (i) N is commutative, (ii) for every x in R there exists x′ϵ such that x−x2x′ϵN, where denotes the subring generated by x, (iii) for every x,y in R, there exists an integer n=n(x,y)≥1 such that both (xy)n−(yx)n and (xy)n