RINGS WITH ASSOCIATORS IN THE LEFT AND MIDDLE NUCLEUS

Authors

  • CHEN-TE YEN Department of Mathematics, Chung Yuan Christian University, Chung Li, Taiwan, 320, Re­public of China.

DOI:

https://doi.org/10.5556/j.tkjm.23.1992.4560

Keywords:

Nonassociative ring, semiprirne ring, prime ring, simple ring

Abstract

Let $R$ be a nonassociative ring, $N$, $M$ and $L$ the left, middle and right nucleus respectively. It is shown that if $R$ a semipnme ring satisfying $(R,R,R) \subset N\cap M$ (resp. $(R,R,R) \subset M\cap L$), then $L\subset M\subset N$(resp. $N\subset M\subset L$); moreover, $R$ is associative if $((R,R,M),(R,R,R)) = 0$ (resp. $((M,R,R),(R,R,R)) = 0)$ or $(M,R) \subset M$; and the Abelian group $(R,+ )$ has no elements of order 2. We also prove that if $R$ is a simple ring satisfying char $R \neq 2$, and $(R, R, R) \subset N \cap M$ or $(R, R, R) \subset M \cap L$ then $R$ is associative.

References

E. Kleinfeld, "A class of rings which are very nearly associative", Amer. Math. Monthly, 93 (1986), 720-722.

E. Kleinfeld, "Rings with (x, y, z) and commutators in the left nucleus", Comm. in Al­gebra, 16 (1988), 2023-2029.

A. Thedy, "On rings with commutators in the nuclei", Math. Z., 119 (1971), 213-218.

C. T. Yen, "Rings with (x,y,z)+(z,y,x) and (N+R^2,R) in the left nucleus", unpublished manuscript.

C. T. Yen, "Rings with (x, R,x) and (N + NR,R) in the left nucleus", Tamkang J. Math., 23 (1992), 247-251.

C. T. Yen, "Rings with (x, y, z) + (z, y, x) and (N, R) in the left nucleus", submitted.

C. T. Yen, "Rings with (x, y, z) + (z, y, x), (N, R) and ((R, R), R, R) in the left nucleus" submitted.

C. T. Yen, "Rings with associators in the left and right nucleus", submitted.

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Published

1992-12-01

How to Cite

YEN, C.-T. (1992). RINGS WITH ASSOCIATORS IN THE LEFT AND MIDDLE NUCLEUS. Tamkang Journal of Mathematics, 23(4), 363–369. https://doi.org/10.5556/j.tkjm.23.1992.4560

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Section

Papers