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Document Type |
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Article In Journal |
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Document Title |
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Commutativity of rings with constraints تبديليه من الحلقات مع القيود |
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Subject |
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Algebra-Ring Theory |
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Document Language |
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English |
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Abstract |
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Suppose that $R$ is an associative ring with identity $1$, $J(R)$ the Jacobson radical of $R$, and $N(R)$ the set of nilpotent elements of $R$. Let $m \geq1$ be a fixed positive integer and $R$ an $m$-torsion-free ring with identity $1$. The main result of the present paper asserts that $R$ is commutative if $R$ satisfies both the conditions \item{(i)} $[x^m,y^m] = 0$ for all $x,y \in R \setminus J(R)$ and \item{(ii)} $[(xy)^m + y^mx^m, x] = 0 = [(yx)^m + x^my^m, x]$, for all $x,y \in R \setminus J(R)$. This result is also valid if (i) and (ii) are replaced by (i)$'$ $[x^m,y^m] = 0$ for all $x,y \in R \setminus N(R)$ and (ii)$'$ $[(xy)^m + y^m x^m, x] = 0 = [(yx)^m + x^m y^m, x]$ for all $x,y \in R\backslash N(R) $. Other similar commutativity theorems are also discussed. |
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ISSN |
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1319-0989 |
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Journal Name |
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Arts and Humanities Journal |
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Volume |
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53 |
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Issue Number |
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3 |
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Publishing Year |
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1424 AH
2003 AD |
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Article Type |
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Article |
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Added Date |
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Wednesday, January 12, 2011 |
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Researchers
| محرم على خان | Khan, Moharram Ali | Researcher | Doctorate | mkhan91@gmail.com |
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