ON THE COMMUTATIVITY AND ANTICOMMUTATIVITY OF RINGS II

Authors

  • CHEN-TE YEN Department of Mathematics, Chung Yuan Christian University, Chung Li, Taiwan, 32023, Republic of China.

DOI:

https://doi.org/10.5556/j.tkjm.22.1991.4610

Keywords:

Jacobson radical, nil ideal, division, primitive, semisimple, semiprime ring

Abstract

It is shown that if $R$ is any associative ring such that for each $x,y\in R$, there exist an even natural number $m(x,y)$ and an odd natural number $n(x,y)$, depending on $x$ and $y$, with either $[x,y]^{m(x,y)} = [x,y]^{n(x,y)}$ or $(x\circ y)^{m(x, y)} = (x \circ y)^{n(x,y)}$, then either $[x,y]$ or $(x \circ y)$ is nilpotent for all $x, y$ in $R$. Moreover, $R$ is commutative if $R$ has no nonzero nil right ideals.

References

I. N. Herstein, "A condition for the commutativity of rings", Canad. J. Math., 9, 583-586, 1957.

I. N. Herstein, "Noncommutative rings", Carus Mathematical Monographs, No. 15, Mathematical Association of America, Washington, DC, 1968.

D. Machale, "An anticommutativity consequence of a ring commutativity theorem of Herstein", Amer. Math. Monthly, 94, 162-165, 1987.

C. T. Yen, "On the commutativity and anticommutativity of rings", Chinese J. Math., 15, 231-235, 1987.

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Published

1991-09-01

How to Cite

YEN, C.-T. (1991). ON THE COMMUTATIVITY AND ANTICOMMUTATIVITY OF RINGS II. Tamkang Journal of Mathematics, 22(3), 267–270. https://doi.org/10.5556/j.tkjm.22.1991.4610

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Papers