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Classification of rings satisfying some constraints on subsets

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*
Moharram A. Khan
*

** Address.**
Centre for Interdisciplinary Research in Basic Sciences (CIRBSc)

Jamia Millia Islamia, Jamia Nagar, New Delhi -110025, India

** E-mail:**
moharram_a@yahoo.com

**Abstract.**
Let $R$ be an associative ring with identity $1$ and $J(R)$ the
Jacobson radical of $R$. Suppose that $m\geq 1$ is a fixed positive integer
and $R$ an $m$-torsion-free ring with $1$. In the present paper, it is shown
that $R$ is commutative if $R$ satisfies both the conditions (i) $[x^m,y^m]=0$
for all $x,y\in R\backslash J(R)$ and (ii) $[x,[x,y^m]]=0$, for all
$x,y\in R\backslash J(R)$. This result is also valid if (ii) is replaced by
(ii)' $[(yx)^mx^m-x^m(xy)^m,x]=0$, for all $x,y\in R\backslash N(R)$.
Our results generalize many well-known commutativity theorems
(cf.\ [1], [2], [3], [4], [5], [6], [9], [10], [11] and [14]).

**AMSclassification. ** 16U80, 16D70.

**Keywords. ** Jacobson radical, nil commutator, periodic ring.