# Article

 Title: Classification of rings satisfying some constraints on subsets (English) Author: Khan, Moharram A. Language: English Journal: Archivum Mathematicum ISSN: 0044-8753 (print) ISSN: 1212-5059 (online) Volume: 43 Issue: 1 Year: 2007 Pages: 19-29 Summary lang: English . Category: math . Summary: Let $R$ be an associative ring with identity $1$ and $J(R)$ the Jacobson radical of $R$. Suppose that $m\ge 1$ 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]). (English) Keyword: Jacobson radical Keyword: nil commutator Keyword: periodic ring MSC: 16U80 idZBL: Zbl 1156.16304 idMR: MR2310121 . Date available: 2008-06-06T22:50:21Z Last updated: 2012-05-10 Stable URL: http://hdl.handle.net/10338.dmlcz/108046 . Reference: [1] Abu-Khuzam H., Tominaga H., Yaqub A.: Commutativity theorems for $s$-unital rings satisfying polynomial identities.Math. J. Okayama Univ. 22 (1980), 111–114. Zbl 0451.16023, MR 0595791 Reference: [2] Abu-Khuzam H.: A commutativity theorem for periodic rings.Math. Japon. 32 (1987), 1–3. Zbl 0609.16020, MR 0886192 Reference: [3] Abu-Khuzam H., Bell H. E., Yaqub A.: Commutativity of rings satisfying certain polynomial identities.Bull. Austral. Math. Soc. 44 (1991), 63–69. Zbl 0721.16020, MR 1120394 Reference: [4] Abu-Khuzam H., Yaqub A.: Commutativity of rings satisfying some polynomial constraints.Acta Math. Hungar. 67 (1995), 207–217. MR 1315805 Reference: [5] Bell H. E.: Some commutativity results for periodic rings.Acta Math. Acad. Sci. Hungar. 28 (1976), 279–283. Zbl 0335.16035, MR 0419535 Reference: [6] Bell H. E.: On rings with commutativity powers.Math. Japon. 24 (1979), 473–478. MR 0557482 Reference: [7] Herstein I. N.: A note on rings with central nilpotent elements.Proc. Amer. Math. Soc. 5 (1954), 620. Zbl 0055.26003, MR 0062714 Reference: [8] Herstein I. N.: A commutativity theorem.J. Algebra 38 (1976), 112–118. Zbl 0323.16014, MR 0396687 Reference: [9] Herstein I. N.: Power maps in rings.Michigan Math. J. 8 (1961), 29–32. Zbl 0096.25701, MR 0118741 Reference: [10] Hirano Y., Hongon M., Tominaga H.: Commutativity theorems for certain rings.Math. J. Okayama Univ. 22 (1980), 65–72. MR 0573674 Reference: [11] Hongan M., Tominaga H.: Some commutativity theorems for semiprime rings.Hokkaido Math. J. 10 (1981), 271–277. MR 0662304 Reference: [12] Jacobson N.: Structure of Rings.Amer. Math. Soc. Colloq. Publ. Providence 1964. Reference: [13] Kezlan T. P.: A note on commutativity of semiprime $PI$-rings.Math. Japon. 27 (1982), 267–268. Zbl 0481.16013, MR 0655230 Reference: [14] Khan M. A.: Commutativity of rings with constraints involving a subset.Czechoslovak Math. J. 53 (2003), 545–559. Zbl 1080.16508, MR 2000052 Reference: [15] Nicholson W. K., Yaqub A.: A commutativity theorem for rings and groups.Canad. Math. Bull. 22 (1979), 419–423. Zbl 0605.16020, MR 0563755 .

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