NONASSOCIATIVE RINGS WITH A SPECIAL DERIVATION

Authors

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

DOI:

https://doi.org/10.5556/j.tkjm.26.1995.4397

Keywords:

Nonassociative ring, nucleus, derivation, semiprime ring, prime ring, simple ring, d-invariant

Abstract

Let $R$ be a nonassociative ring, $N$, $L$ and $G$ the left nucleus, right nucleus and nucleus respectively. It is shown that if $R$ is a prime ring with a derivafion $d$ such that $ax+ d(x) \in G$ where $a \in Z$, the ring of rational integers, or $a \in G$ with $(ad)(x) = ad(x) = d(ax)$ and $ax = xa$ for all $x$ in $R$ then either $R$ is associative or $ad+ d^2 = 2d(R)^2 = 0$. This result is also valid under the weaker hypothesis $ax+ d(x) \in N \cap L$ for all $x$ in $R$ for the simple ring case, and we obtain that either $R$ is associative or $((ad+ d^2)(R))^2 =0$ for the prime ring case.

References

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

P.H . Lee and T . K. Lee, "Note on nilpotent derivations," Proc. Amer. Math. Soc., 98(1986), 31-32.

T. I. Suh, "Prime nonassociative rings with a special derivation," Abstracts of papers presented to the Amer. Math. Soc., 14(1993), 284.

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

C. T. Yen, "Rings with associators in the nuclei," unpublished manuscript.

C. T. Yen, "Rings with a derivation whose image is contained in the nuclei," Tamkang J. Math., 25(1994), 301-307.

C. T. Yen, "Rings with a Jordan derivation whose image is contained in the nuclei or commutative center," submitted.

C. T. Yen, "Prime ring with a derivation whose some power image is contained in the nucleus," Soochow J. Math. to appear.

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Published

1995-09-01

How to Cite

YEN, C.-T. (1995). NONASSOCIATIVE RINGS WITH A SPECIAL DERIVATION. Tamkang Journal of Mathematics, 26(3), 193-199. https://doi.org/10.5556/j.tkjm.26.1995.4397

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