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 Commutativity of one sideds-unital rings through a Streb's result

1997 ◽  
Vol 20 (2) ◽  
pp. 267-270
Author(s):  
Murtaza A. Quadri ◽  
V. W. Jacob ◽  
M. Ashraf
Keyword(s):  
Unital Ring ◽  
Commutativity Theorems ◽  
Ring Elements ◽  
Unital Rings

The main theorem proved in the present paper states as follows “Letm,k,nandsbe fixed non-negative integers such thatkandnare not simultaneously equal to1andRbe a left (resp right)s-unital ring satisfying[(xmyk)n−xsy,x]=0(resp[(xmyk)n−yxs,x]=0) ThenRis commutative.” Further commutativity of lefts-unital rings satisfying the conditionxt[xm,y]−yr[x,f(y)]xs=0wheref(t)∈t2Z[t]andm>0,t,randsare fixed non-negative integers, has been investigated Finally, we extend these results to the case when integral exponents in the underlying conditions are no longer fixed, rather they depend on the pair of ring elementsxandyfor their values. These results generalize a number of commutativity theorems established recently.

scholarly journals SOME COMMUTATIVITY THEOREMS FOR ASSOCIATIVE RINGS WITH CONSTRAINTS INVOLVING A NIL SUDSET

1991 ◽  
Vol 22 (3) ◽  
pp. 285-297
Author(s):  
MOHD ASHRAF
Keyword(s):  
Related Result ◽  
Commutativity Theorems ◽  
Associative Rings ◽  
Unital Rings

We first prove that a ring $R$ with unity 1 is corrunutalive if and only if for each $x$ in $R$ either $x$ is central or there exists a polynomial $f(t) \in Z[t]$ such that $x- x^2f(x) \in A$, where $A$ is a nil subset of $R$ (not necessarily a subring of $R$) and $R$ stisfies any one of the conditions $[x, x^my- x^py^nx^q] =0$ and $[x,yx^m-x^Py^nx^q]=0$ for all $x,y$ in $R$, where $m\ge 0$, $n >1$, $p \ge 0$, $q \ge 0$ are integers depending on pair of elements $x$, $y$. Further the same result has been extended for one sided $s$-unital rings. Finally a related result for a nil commutative subset $A$ is also obtained.

scholarly journals On commutativity of one-sideds-unital rings

1992 ◽  
Vol 15 (4) ◽  
pp. 813-818
Author(s):  
H. A. S. Abujabal ◽  
M. A. Khan
Keyword(s):  
Polynomial Identity ◽  
Unital Ring ◽  
Unital Rings

The following theorem is proved: Letr=r(y)>1,s, andtbe non-negative integers. IfRis a lefts-unital ring satisfies the polynomial identity[xy−xsyrxt,x]=0for everyx,y∈R, thenRis commutative. The commutativity of a rights-unital ring satisfying the polynomial identity[xy−yrxt,x]=0for allx,y∈R, is also proved.

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 Commutativity theorems for rings with polynomial constraints on certain subsets

1991 ◽  
Vol 43 (3) ◽  
pp. 451-462 ◽  
Author(s):  
Hiroaki Komatsu ◽  
Hisao Tominaga
Keyword(s):  
Commutativity Theorems ◽  
Unital Rings ◽  
Polynomial Constraints

We prove several commutativity theorems for unital rings with polynomial constraints on certain subsets, which improve and generalise the recent results of Grosen, and Ashraf and Quadri.

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.

Commutativity theorems for s-unital rings with constraints on commutators

1992 ◽  
Vol 21 (3-4) ◽  
pp. 256-263
Author(s):  
Hamza A. S. Abujabal ◽  
Veselin Peric
Keyword(s):  
Commutativity Theorems ◽  
Unital Rings

scholarly journals A commutativity theorem for power-associative rings

1970 ◽  
Vol 3 (1) ◽  
pp. 75-79 ◽  
Author(s):  
D. L. Outcalt ◽  
Adil Yaqub
Keyword(s):  
Finite Field ◽  
Associative Ring ◽  
Commutativity Theorem ◽  
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 x2 = y2 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 “x2 = y2” is replaced by “xk = yk” for any integer k > 2.

scholarly journals COMMUTATIVITY THEOREMS FOR RINGS WITH CONSTRAINTS ON COMMUTATORS

1995 ◽  
Vol 26 (1) ◽  
pp. 25-29
Author(s):  
H. A. S. ABUJABAL ◽  
MOHD ASHRAF
Keyword(s):  
Positive Integer ◽  
Unital Ring ◽  
Commutativity Theorems

Let $R$ be a left (resp. right) $s$-unital ring and $m$ be a positive integer. Suppose that for each $y$ in $R$ there exist $J(t)$, $g(t)$, $h(t)$ in $Z[t]$ such that $x^m[x,y]= g(y)[x,y^2f(y)]h(y)$ (resp. $[x,y]x^m= g(y)[x,y^2f(y)]h(y))$ for all $x$ in $R$. Then $R$ is commutative (and conversely). Finally, the result is extended to the case when the exponent $m$ depends on the choice of $x$ and $y$.

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