Commutativity of Rings with Constraints Involving a Subset

Author: Khan M.A.

Source: Czechoslovak Mathematical Journal, Volume 53, Number 3, September 2003 , pp. 545-559(15)

Publisher: Springer

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Abstract:

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

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 <IMG SRC="http://images.ingentaselect.com/absimages/klu/00114642/klu_cmaj_2003_53_3_426226h.1.gif" ALT="x,\ y\in R \backslash\thinspace J(R)" TEXT="a mathematical formula"> and

(ii) [(xy)^m + y^mx^m, x] = 0 = [(yx)^m + x^my^m, x], for all <IMG SRC="http://images.ingentaselect.com/absimages/klu/00114642/klu_cmaj_2003_53_3_426226h.2.gif" ALT="x,\ y\in R \backslash\thinspace J(R)." TEXT="a mathematical formula">

This result is also valid if (i) and (ii) are replaced by (i) prime

[x^m, y^m] = 0 for all <IMG SRC="http://images.ingentaselect.com/absimages/klu/00114642/klu_cmaj_2003_53_3_426226h.1.gif" ALT="x,\ y\in R \backslash\thinspace J(R)" TEXT="a mathematical formula"> and (ii) prime

[(xy)^m + y^mx^m, x] = 0 = [(yx)^m + x^my^m, x] for all <IMG SRC="http://images.ingentaselect.com/absimages/klu/00114642/klu_cmaj_2003_53_3_426226h.2.gif" ALT="x,\ y\in R \backslash\thinspace J(R)." TEXT="a mathematical formula">

Other similar commutativity theorems are also discussed.

Keywords: commutativity theorems; Jacobson radicals; nilpotent elements; periodic rings; torsion-free rings

Document Type: Research article

DOI: http://dx.doi.org/10.1023/B:CMAJ.0000024502.32510.e5

Affiliations: 1: Department of Mathematics, King Abdulaziz University, P.O. Box 30356, Jeddah-21477, Saudi Arabia, Email: nassb@hotmail.com

Publication date: 2003-09-01