Structure of weakly periodic rings with potent extended commutators

Subperiodic Rings with Conditions on Extended Commutators

Let R be a ring with Jacobson radical J and with center C. Let P be the set of potent elements x for which xk = x for some integer k > 1. Let N be the set of nilpotents. A ring R is called subperiodic if R \ (J ∪ C) ⊆ N + P . We consider the commutativity behavior of a subperiodic ring with some constraint involving extended commutators. Mathematics Subject Classification: 16U80, 16D70

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.

WEAKLY g(x)-CLEAN RINGS

A ring $R$ with identity is called ``clean'' if $~$for every element $ain R$, there exist an idempotent $e$ and a unit $u$ in $R$ such that $a=u+e$. Let $C(R)$ denote the center of a ring $R$ and $g(x)$ be a polynomial in $C(R)[x]$. An element $rin R$ is called ``g(x)-clean'' if $r=u+s$ where $g(s)=0$ and $u$ is a unit of $R$ and, $R$ is $g(x)$-clean if every element is $g(x)$-clean. In this pa...

Vanishing Power Values of Commutators with Derivations on Prime Rings

WEAKLY PERIODIC AND SUBWEAKLY PERIODIC RINGS AMBER ROSIN and ADIL YAQUB

Throughout, R represents a ring and (R) denotes the commutator ideal of R. For any x,y in R, [x,y] = xy −yx is the usual commutator. A word w(x,y) is a product in which each factor is x or y . The empty word is defined to be 1. We now state formally the definition of a subweakly periodic ring. Definition 1. A ring R is called subweakly periodic if every x in R\J can be written in the form x = a...