Goldie's theorem for alternative rings

Authors:
H. Essannouni and A. Kaidi

Journal:
Proc. Amer. Math. Soc. **121** (1994), 39-45

MSC:
Primary 17D05

MathSciNet review:
1181162

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Abstract | References | Similar Articles | Additional Information

Abstract: It is known that the socle of a semiprime Goldie ring is generated by a central idempotent and that a prime Goldie ring with a nonzero socle is a simple artinian ring. We prove the extension of these results to alternative rings. We also give an analogue of Goldie's theorem for alternative rings. A Goldie-like theorem was obtained earlier by the authors for noetherian alternative rings by a quite different method.

**[1]**R. H. Bruck and Erwin Kleinfeld,*The structure of alternative division rings*, Proc. Amer. Math. Soc.**2**(1951), 878–890. MR**0045099**, 10.1090/S0002-9939-1951-0045099-9**[2]**A. W. Chatters and C. R. Hajarnavis,*Rings with chain conditions*, Research Notes in Mathematics, vol. 44, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1980. MR**590045****[3]**Hassane Essannouni and Amin Kaidi,*Semiprime alternative rings with a.c.c*, Ring theory (Granada, 1986) Lecture Notes in Math., vol. 1328, Springer, Berlin, 1988, pp. 82–93. MR**959747**, 10.1007/BFb0100918**[4]**M. Slater,*Ideals in semiprime alternative rings*, J. Algebra**8**(1968), 60–76. MR**0233862****[5]**M. Slater,*Alternative rings with 𝑑.𝑐.𝑐. I*, J. Algebra**11**(1969), 102–110. MR**0233863****[6]**M. Slater,*The socle of an alternative ring*, J. Algebra**14**(1970), 443–463. MR**0260817****[7]**K. A. Zhevlakov, A. M. Slin′ko, I. P. Shestakov, and A. I. Shirshov,*Rings that are nearly associative*, Pure and Applied Mathematics, vol. 104, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982. Translated from the Russian by Harry F. Smith. MR**668355**

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1994-1181162-5

Keywords:
Nonassociative ring,
alternative ring,
nucleus,
center,
socle,
Goldie ring,
semiprime alternative ring,
prime alternative ring,
essential right ideal,
regular element

Article copyright:
© Copyright 1994
American Mathematical Society