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 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 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.

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 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 COMMUTATIVITY OF RIGHT $S$-UNITAL RINGS UNDER SOME POLYNOMIAL CONSTRAINTS

1993 ◽  
Vol 24 (1) ◽  
pp. 23-28
Author(s):  
M. ASHRAF ◽  
M. A. QUADRI ◽  
V. W. JACOB
Keyword(s):  
Ring Elements ◽  
Unital Rings ◽  
Polynomial Constraints

In the present paper we discuss the commutativity of certain rings, namely rings with unity 1 and right s-unital rings under each of the following conditions: \[ (P1)[yx^m - x^nf (y), x] = 0, \quad (P1)^*[yx^m - f (y)x^n, x] = 0, \] where $m$, $n$ are fixed non-negative integers and $f(x)$ is a polynomial in $X^2\mathbb{Z}(X)$ varying with the pair of ring elements $x$, $y$. Further, the results have been extended to the case when $m$ and $n$ depend on the choice of $x$ and $y$ and the ring satisfies the Chacron's condition.

Some Commutativity Theorems for Left s-Unital Rings

1989 ◽  
Vol 15 (3-4) ◽  
pp. 335-342 ◽  
Author(s):  
Hiroaki Komatsu ◽  
Hisao Tominaga
Keyword(s):  
Commutativity Theorems ◽  
Unital Rings

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.

Commutativity for a Certain Class of Rings

1998 ◽  
Vol 5 (4) ◽  
pp. 301-314
Author(s):  
H. A. S. Abujabal ◽  
M. A. Khan
Keyword(s):  
Semiprime Ring ◽  
Commutativity Theorems ◽  
Unital Rings

Abstract We first establish the commutativity for the semiprime ring satisfying [xn, y]xr = ±ys [x, ym ]yt for all x, y in R, where m, n, r, s and t are fixed non-negative integers, and further, we investigate the commutativity of rings with unity under some additional hypotheses. Moreover, it is also shown that the above result is true for s-unital rings. Also, we provide some counterexamples which show that the hypotheses of our theorems are not altogether superfluous. The results of this paper generalize some of the well-known commutativity theorems for rings which are right s-unital.

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.

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