A commutativity theorem

Throughout, R is a ring with set of nilpotents N and center C. The commutator ideal is denoted by C(R). As usual, [x, y] denotes the commutator xy -yx. In [2], it was proved that if n is a fixed positive integer, and R is an n(n + 1)-torsion-free periodic ring such that (xy) ~ -y~x ~ E C and (xy) TM -yn+lxn+l E C, and if N is commutative, then R is commutative. We generalize this theorem by deleting the condition (xy) TM -yn+lxn+l E E C and establishing the commutativity of R under the weaker conditions. We also generalize Theorem A below which was proved in [3]. THEOREM A. Let R be a ring with identity and let n be a fixed positive integer. Suppose that R is n-torsion-free, and that for all x, y in R, [x n, yn] = = 0 and (xy) n+l -xn+ly n+l is in the centerC of R. Then R is commutative.We generalize Theorem A in two ways. First, we assume that the two conditions in Theorem A hold merely for all x E R\N, y E R\N. Secondly, we assume that these two conditions hold for all x E R\J, y E R\J instead, where J denotes the Jacobson radical of R. In both situations, we establish that R is commutative (under these weaker conditions). We also give examples which show that neither of these two theorems need be true if any one of the hypotheses is deleted.We begin with the following well known lemma. LEMMA 1. If Ix, Y] commutes with x, then [xk,y] = kxk-l[x,y] for allpositive integers k. LEMMA 2. If the commutator ideal, C(R), of R is nil, then the nilpotents N of R form an ideal. PROOF. Suppose a E N, b E N. Let R = R/C(R). Then -d = a + C(R)and -b = b + C(R) are also nilpotent. Since R is commutative, therefore -b is nilpotent, say (g-~)k = g, and hence (a-b) k E C(R) C_ N, by hypothesis. Thus, a -b E N. Now, suppose x is an arbitrary element of R. Then a x is also nilpotent, say (gg)m = ~, which implies (ax) m E C(R) C= N, and thus ax E N. Similarly, xa E N for all a E N, x E R. Hence, N is an ideal of R.We start with Theorems i and 2 which generalize Theorem A above. Then we prove Theorem 3 which generalizes the results of [2].

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“…Subtracting and keeping in mind that N 2 C C, we get By a lemma in [7], (45) and (46)imply that (47) [[a,b],b] -= 0 for all a 9 N, b 9 R.…”

Section: [(A+l)b]n-b"(a+l)nec and [B(a+l)]n-(a+l)%"ecmentioning

confidence: 92%

Commutativity of rings satisfying some polynomial conditions

Abu-Khuzam

Yaqub

1995

Acta Math Hung

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“…In preparation for the proof of Theorem 1, we state the following known lemmas [2,11,5]. PROOF OF THEOREM 1: By hypothesis, [x n ,y n ] = 0 for all x,y in R and hence, by [8], the commutator ideal is nil.…”

Section: Suppose That R Is N-torsion-free and That For All Xy In Rmentioning

confidence: 99%

Commutativity of rings satisfying certain polynomial identities

Abu-Khuzam

Bell

Yaqub

1991

Bull. Austral. Math. Soc.

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It is shown that an n-torsion-free ring R with identity such that, for all x, y in R, xnyn = ynyn and (xy)n+1 − xn+1yn+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.

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“…By subtracting the two expressions in (9) and (10), and using the facts in (8), it can be seen that [a,…”

Section: Lemma 3([4])mentioning

confidence: 99%

On weakly periodic-like rings and commutativity theorems

Abu-Khuzam¹,

Bell²,

Yaqub³

2006

Tamkang J. Math.

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A ring $R$ is called periodic if, for every $x$ in $R$, there exist distinct positive integers $m$ and $n$ such that $x^m=x^n$. An element $x$ of $R$ is called potent if $x^k=x$ for some integer $k>1$. A ring $R$ is called weakly periodic if every $x$ in $R$ can be written in the form $x=a+b$ for some nilpotent element $a$ and some potent element $b$ in $R$. A ring $R$ is called weakly periodic-like if every element $x$ in $R$ which is not in the center $C$ of $R$ can be written in the form $x=a+b$, with $a$ nilpotent and $b$ potent. Some structure and commutativity theorems are established for weakly periodic-like rings $R$ satisfying certain torsion-freeness hypotheses along with conditions involving some elements being central.

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A commutativity theorem

Cited by 11 publications

References 2 publications

Commutativity of rings satisfying some polynomial conditions

Commutativity of rings satisfying some polynomial conditions

Commutativity of rings satisfying certain polynomial identities

On weakly periodic-like rings and commutativity theorems

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