RINGSWITH (R,R,R)AND [R,(R,R,R)] IN THE LEFT NUCLEUS

Authors

  • IRVIN ROY HENTZEL Department Of Mathematics, 400 Carver Hall, Iowa State University, Ames, Iowa, 50011, U.S.A
  • CHEN-TE YEN Department Of Mathematics, Chung Yuan University, Chung Li, Taiwan, 320, Republic Of China.

DOI:

https://doi.org/10.5556/j.tkjm.27.1996.4358

Keywords:

Nonassociative ring, semiprime ring, associator ideal

Abstract

Let $R$ be a nonassociative ring, and $N = (R,R,R) + [R,(R,R,R)]$. We show that $W = \{w\in N | Rw +wR +R(wR) \subset N\}$ is a two-sided ideal of $R$. If for some $r\in R$, any one of the sets $(r, R, R)$, $(R,r, R)$ or $(R, R,r)$ is contained in $W$, then the other two sets are contained in $W$ also. If the associators are assumed to be contained in either the left, the middle, or the right nucleus, and $I$ is the ideal generated by all associators, then $I^2 \subset W$. If $N$ is assumed to be contained in the left or the right nucleus, then $W^2 = 0$. We conclude that if $R$ is semiprime and $N$ is contained in the left or the right nucleus, then $R$ is associative. We assume characteristic not 2.

References

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

Chen-Te Yen, "Rings with associators in the left and middle nucleus," Tamkang J. Math., 23(1992), 363-369.

一 , "Rings with (R,R,R) and ((R,R,R),R) in the left nucleus," Tamkang J. Math., 24(1993), 209-213.

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Published

1996-06-01

How to Cite

HENTZEL, I. R., & YEN, C.-T. (1996). RINGSWITH (R,R,R)AND [R,(R,R,R)] IN THE LEFT NUCLEUS. Tamkang Journal of Mathematics, 27(2), 183–187. https://doi.org/10.5556/j.tkjm.27.1996.4358

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Papers