scholarly journals ON COMMUTATIVITY THEOREMS FOR P. I. - RINGS WITH UNITY

1993 ◽  
Vol 24 (1) ◽  
pp. 29-36
Author(s):  
THOMAS P. KEZLAN
Keyword(s):  
Commutativity Theorem ◽  
Commutativity Theorems

The purpose of this paper is to show how a previous commutativity theorem for general rings can be used to prove commutativity theorems for rings with unity, and to obtain several new results via this route, e.g., if a ring with unity satisfies either $x^k[x^n, y] = [x, y^m]x^\ell$ or $x^k[x^n,y] = [x,y^m]y^\ell (m > 1)$ and if either (A) $m$ and $n$ are relatively prime or (B) $n[x,y]=0$ implies $[x,y]=0$, then $R$ is commutative.

scholarly journals A commutativity theorem for lefts-unital rings

1990 ◽  
Vol 13 (4) ◽  
pp. 769-774
Author(s):  
Hamza A. S. Abujabal
Keyword(s):  
Commutativity Theorem ◽  
Unital Ring ◽  
Commutativity Theorems ◽  
Associative Rings ◽  
Unital Rings

In this paper we generalize some well-known commutativity theorems for associative rings as follows: LetRbe a lefts-unital ring. If there exist nonnegative integersm>1,k≥0, andn≥0such that for anyx,yinR,[xky−xnym,x]=0, thenRis commutative.

scholarly journals An iteration technique and commutativity of rings

1990 ◽  
Vol 13 (2) ◽  
pp. 315-319
Author(s):  
H. A. S. Abujabal ◽  
M. S. Khan
Keyword(s):  
Commutativity Theorems ◽  
Iteration Technique

Through much shorter proofs, some new commutativity theorems for rings with unity have been obtained. These results either extend or generalize a few well-known theorems. Our method of proof is based on an iteration technique.

scholarly journals ON COMMUTATIVITY OF ONE-SIDED $s$-UNITAL RINGS

1992 ◽  
Vol 23 (3) ◽  
pp. 253-268
Author(s):  
H. A. S. ABUJABAL ◽  
M. A. KHAN ◽  
M. S. SAMMAN
Keyword(s):  
Commutativity Theorems ◽  
Unital Rings

In the present paper, we study the commutativity of one sided s-unital rings satisfying conditions of the form $[x^r y\pm x^ny^mx^s,x]= 0 = [x^ry^m\pm x^ny^{m^2}x^s, x]$, or $[yx^r\pm x^ny^mx^s, x] = 0 = [y^mx^r\pm x^ny^{m^2}x^s, x]$ for each $x$,$y \in R$, where $m = m(y) > 1$ is an integer depending on $y$ and $n$, $r$ and $s$ are fixed non-negative integers. Other commutativity theorems are also obtained. Our results generalize·some of the well-known commutativity theorems for rings.

scholarly journals A Commutativity theorem for semiprime rings

1980 ◽  
Vol 30 (1) ◽  
pp. 33-36
Author(s):  
Vishnu Gupta
Keyword(s):  
Positive Integer ◽  
Semiprime Ring ◽  
Commutativity Theorem ◽  
Semiprime Rings

AbstractIt is shown that if R is a semiprime ring with 1 satisfying the property that, for each x, y ∈ R, there exists a positive integer n depending on x and y such that (xy)k − xkyk is central for k = n,n+1, n+2, then R is commutative, thus generalizing a result of Kaya.

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