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Keywords:
commutators; division rings; factorsubrings; polynomial identities; torsion-free rings
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.
References:
[1] H. E. Bell, M. A. Quadri and M. A. Khan: Two commutativity theorems for rings. Rad. Mat. 3 (1987), 255–260. MR 0931981
[2] M. Chacron: A commutativity theorem for rings. Proc. Amer. Math. Soc. 59 (1976), 211–216. DOI 10.1090/S0002-9939-1976-0414636-1 | MR 0414636 | Zbl 0341.16020
[3] I. N. Herstein: Two remarks on commutativity of rings. Canad. J. Math. 7 (1955), 411–412. DOI 10.4153/CJM-1955-044-2 | MR 0071405
[4] T. P. Kezlan: A note on commutativity of semiprime PI-rings. Math. Japon. 27 (1982)), 267–268. MR 0655230 | Zbl 0481.16013
[5] M. A. Khan: Commutativity of right $s$-unital rings with polynomial constraints. J. Inst. Math. Comput. Sci. 12 (1999), 47–51. MR 1693433 | Zbl 0935.16023
[6] H. Komatsu and H. Tominaga: Chacron’s condition and commutativity theorems. Math. J. Okayama Univ. 31 (1989), 101–120. MR 1043353
[7] E. Psomopoulos: Commutativity theorems for rings and groups with constraints on commutators. Internat. J. Math. Math. Sci. 7 (1984), 513–517. DOI 10.1155/S0161171284000569 | MR 0771600 | Zbl 0561.16013
[8] M. O. Searoid and D. MacHale: Two elementary generalisations of Boolean rings. Amer. Math. Monthly, 93 (1986), 121–122. DOI 10.2307/2322707 | MR 0827587
[9] W. Streb: Zur Struktur nichtkommutativer Ringe. Math. J. Okayama Univ. 31 (1989), 135–140. MR 1043356 | Zbl 0702.16022
[10] H. Tominaga and A. Yaqub: Commutativity theorems for rings with constraints involving a commutative subset. Results Math. 11 (1987), 186–192. DOI 10.1007/BF03323267 | MR 0880201
[11] J. Tong: On the commutativity of a ring with identity. Canad. Math. Bull. 72 (1984), 456–460. DOI 10.4153/CMB-1984-071-x | MR 0763045 | Zbl 0545.16015

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