Geodesic
Commutativity of rings with constraints involving a subset
Czechoslovak Mathematical Journal, Tome 53 (2003) no. 3, pp. 545-559
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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 $m \ge 1$ 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 (i) $[x^m,y^m] = 0$ for all $x,y \in R \setminus J(R)$ and (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)$^{\prime }$ $[x^m,y^m] = 0$ for all $x,y \in R \setminus N(R)$ and (ii)$^{\prime }$ $[(xy)^m + y^m x^m, x] = 0 = [(yx)^m + x^m y^m, x]$ for all $x,y \in R\backslash N(R) $. Other similar commutativity theorems are also discussed.
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 $m \ge 1$ 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 (i) $[x^m,y^m] = 0$ for all $x,y \in R \setminus J(R)$ and (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)$^{\prime }$ $[x^m,y^m] = 0$ for all $x,y \in R \setminus N(R)$ and (ii)$^{\prime }$ $[(xy)^m + y^m x^m, x] = 0 = [(yx)^m + x^m y^m, x]$ for all $x,y \in R\backslash N(R) $. Other similar commutativity theorems are also discussed.
Classification : 16R50, 16U70, 16U80, 16U99
Keywords: commutativity theorems; Jacobson radicals; nilpotent elements; periodic rings; torsion-free rings 
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     title = {Commutativity of rings with constraints involving a subset},
     journal = {Czechoslovak Mathematical Journal},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2003_53_3_a4/}
}
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Khan, Moharram A. Commutativity of rings with constraints involving a subset. Czechoslovak Mathematical Journal, Tome 53 (2003) no. 3, pp. 545-559. http://geodesic.mathdoc.fr/item/CMJ_2003_53_3_a4/

[1] H. A. S.  Abujabal, H. E. Bell, M. S. Khan and M. A. Khan: Commutativity of semiprime rings with power constraints. Studia Sci. Math. Hungar. 30 (1995), 183–187. | MR

[2] H.  Abu-Khuzam: A commutativity theorem for periodic rings. Math. Japon. 32 (1987), 1–3. | MR | Zbl

[3] H.  Abu-Khuzam and A.  Yaqub: Commutativity of rings satisfying some polynomial constraints. Acta Math. Hungar. 67 (1995), 207–217. | DOI | MR

[4] H.  Abu-Khuzam, H. E.  Bell and A.  Yaqub: Commutativity of rings satisfying certain polynomial identities. Bull. Austral. Math. Soc. 44 (1991), 63–69. | DOI | MR

[5] H. E.  Bell: Some commutativity results for periodic rings. Acta Math. Acad. Sci. Hungar. 28 (1976), 279–283. | DOI | MR | Zbl

[6] H. E.  Bell: A commutativity study for periodic rings. Pacific J.  Math. 70 (1977), 29–36. | DOI | MR | Zbl

[7] H. E.  Bell: On rings with commutativity powers. Math. Japon. 24 (1979), 473–478. | MR

[8] I. N.  Herstein: Power maps in rings. Michigan Math.  J. 8 (1961), 29–32. | DOI | MR | Zbl

[9] I. N.  Herstein: A commutativity theorem. J.  Algebra 38 (1976), 112–118. | DOI | MR | Zbl

[10] Y.  Hirano, M.  Hongon and H.  Tominaga: Commutativity theorems for certain rings. Math.  J. Okayama Univ. 22 (1980), 65–72. | MR

[11] M.  Hongan and H.  Tominaga: Some commutativity theorems for semiprime rings. Hokkaido Math.  J. 10 (1981), 271–277. | MR

[12] N.  Jacobson: Structure of Rings. Amer. Math. Soc. Colloq. Publ., Providence, 1964.

[13] T. P.  Kezlan: A note on commutativity of semiprime PI-rings. Math. Japon 27 (1982), 267–268. | MR | Zbl

[14] W. K.  Nicholson and A.  Yaqub: A commutativity theorem for rings and groups. Canad. Math. Bull. 22 (1979), 419–423. | DOI | MR