Adil Yaqub

A commutativity theorem for rings

Let R be an associative ring with identity in which every element is either nilpotent or a unit. The following results are established. The set N of nilpotent elements in R is an ideal. If R / N is finite and if x ≡ y (mod N ) implies x 2 = y 2 or both x and y commute with all elements of N , then R is commutative. Examples are given to show that R need not be commutative if “X 2 = y 2 ” is replaced by “x k = y K ” for any integer k > 2. The case N = (0) yields Wedderburns Theorem.

Commutativity of rings satisfying certain polynomial identities

It is shown that an n -torsion-free ring R with identity such that, for all x , y in R , x n y n = y n y n and ( xy ) n+1 − x n+1 y n+1 is central, must be commutative. It is also shown that a periodic n –torsion-free ring (not necessarily with identity) for which ( xy ) n − ( yx ) n is always in the centre is commutative provided that the nilpotents of R form a commutative set. Further, examples are given which show that all the hypotheses of both theorems are essential.

Rings with a finite set of nonnilpotents

Let R be a ring and let N denote the set of nilpotent elements of R. Let n be a nonnegative integer. The ring R is called a θn-ring if the number of elements in R which are not in N is at most n. The following theorem is proved: If R is a θn-ring, then R is nil or R is finite. Conversely, if R is a nil ring or a finite ring, then R is a θn-ring for some n. The proof of this theorem uses the structure theory of rings, beginning with the division ring case, followed by the primitive ring case, and then the semisimple ring case. Finally, the general case is considered.

Generalized periodic rings

Let R be a ring, and let N and C denote the set of nilpotents and the center of R, respectively. R is called generalized periodic if for every x∈R\(N⋃C), there exist distinct positive integers m, n of opposite parity such that xn−xm∈N⋂C. We prove that a generalized periodic ring always has the set N of nilpotents forming an ideal in R. We also consider some conditions which imply the commutativity of a generalized periodic ring.

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.

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.

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

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 commutativity theorem for power-associative rings

Let R be a power-associative ring with identity and let I be an ideal of R such that R / I is a finite field and x ≡ y (mod I ) implies x 2 = y 2 or both x and y commute with all elements of I . It is proven that R must then be commutative. Examples are given to show that R need not be commutative if various parts of the hypothesis are dropped or if “ x 2 = y 2 ” is replaced by “ x k = y k ” for any integer k > 2.

CLASSIFICATION OF RINGS SATISFYING SOME HERSTEIN'S CONDITION ON THE SET OF ALL NON-NILPOTENTS

Abstract Let be a non-commutative ring with center and let be the set of all nilpotents of . The following result is proved: If for every , there exists an integer , depending on , such that , then is one of five types whose structures can be determined abstractly.