# Article

 Title: Commutativity of rings through a Streb’s result (English) Author: Khan, Moharram A. Language: English Journal: Czechoslovak Mathematical Journal ISSN: 0011-4642 (print) ISSN: 1572-9141 (online) Volume: 50 Issue: 4 Year: 2000 Pages: 791-801 Summary lang: English . Category: math . Summary: In this paper we investigate commutativity of rings with unity satisfying any one of the properties: \begin{aligned} &\lbrace 1- g(yx^{m}) \rbrace \ [yx^{m} - x^{r} f (yx^{m}) \ x^s, x] \lbrace 1- h (yx^{m}) \rbrace = 0, \\&\lbrace 1- g(yx^{m}) \rbrace \ [x^{m} y - x^{r} f (yx^{m}) x^{s}, x] \lbrace 1- h (yx^{m}) \rbrace = 0, \\&y^{t} [x,y^{n}] = g (x) [f (x), y] h (x)\ {\mathrm and} \ \ [x,y^{n}] \ y^{t} = g (x) [f (x), y] h (x) \end{aligned} for some $f(X)$ in $X^{2} {\mathbb Z}[X]$ and $g(X)$, $h(X)$ in ${\mathbb Z} [X]$, where $m \ge 0$, $r \ge 0$, $s \ge 0$, $n > 0$, $t > 0$ are non-negative integers. We also 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 elements $x$ and $y$ for their values. Further, under different appropriate constraints on commutators, commutativity of rings has been studied. These results generalize a number of commutativity theorems established recently. (English) Keyword: commutators Keyword: division rings Keyword: factorsubrings Keyword: polynomial identities Keyword: torsion-free rings MSC: 16R50 MSC: 16U70 MSC: 16U80 idZBL: Zbl 1079.16504 idMR: MR1792970 . Date available: 2009-09-24T10:37:53Z Last updated: 2016-04-07 Stable URL: http://hdl.handle.net/10338.dmlcz/127610 . Reference: [1] H. E. Bell, M. A. Quadri and M. A. Khan: Two commutativity theorems for rings.Rad. Mat. 3 (1987), 255–260. MR 0931981 Reference: [2] M. Chacron: A commutativity theorem for rings.Proc. Amer. Math. Soc. 59 (1976), 211–216. Zbl 0341.16020, MR 0414636, 10.1090/S0002-9939-1976-0414636-1 Reference: [3] I. N. Herstein: Two remarks on commutativity of rings.Canad. J. Math. 7 (1955), 411–412. MR 0071405, 10.4153/CJM-1955-044-2 Reference: [4] T. P. Kezlan: A note on commutativity of semiprime PI-rings.Math. Japon. 27 (1982)), 267–268. Zbl 0481.16013, MR 0655230 Reference: [5] M. A. Khan: Commutativity of right $s$-unital rings with polynomial constraints.J. Inst. Math. Comput. Sci. 12 (1999), 47–51. Zbl 0935.16023, MR 1693433 Reference: [6] H. Komatsu and H. Tominaga: Chacron’s condition and commutativity theorems.Math. J. Okayama Univ. 31 (1989), 101–120. MR 1043353 Reference: [7] E. Psomopoulos: Commutativity theorems for rings and groups with constraints on commutators.Internat. J. Math. Math. Sci. 7 (1984), 513–517. Zbl 0561.16013, MR 0771600, 10.1155/S0161171284000569 Reference: [8] M. O. Searoid and D. MacHale: Two elementary generalisations of Boolean rings.Amer. Math. Monthly, 93 (1986), 121–122. MR 0827587, 10.2307/2322707 Reference: [9] W. Streb: Zur Struktur nichtkommutativer Ringe.Math. J. Okayama Univ. 31 (1989), 135–140. Zbl 0702.16022, MR 1043356 Reference: [10] H. Tominaga and A. Yaqub: Commutativity theorems for rings with constraints involving a commutative subset.Results Math. 11 (1987), 186–192. MR 0880201, 10.1007/BF03323267 Reference: [11] J. Tong: On the commutativity of a ring with identity.Canad. Math. Bull. 72 (1984), 456–460. Zbl 0545.16015, MR 0763045, 10.4153/CMB-1984-071-x .

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