Prime generalized alternative rings $I$ with nontrivial idempotent
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- by Harry F. Smith
- Proc. Amer. Math. Soc. 39 (1973), 242-246
- DOI: https://doi.org/10.1090/S0002-9939-1973-0313348-X
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Abstract:
A generalized alternative ring $I$ is a nonassociative ring $R$ in which the identities $(wx,y,z) + (w,x,[y,z]) - w(x,y,z) - (w,y,z)x;([w,x],y,z) + (w,x,yz) - y(w,x,z) - (w,x,y)z$; and $(x,x,x)$ are identically zero. It is demonstrated here that if $R$ is a ring of this type with characteristic different from two and three, then $R$ semiprime with idempotent $e$ implies that $R$ has a Peirce decomposition relative to $e$. Furthermore, if $R$ is prime and $e \ne 0,1$; then $R$ must be alternative.References
- A. A. Albert, Power-associative rings, Trans. Amer. Math. Soc. 64 (1948), 552–593. MR 27750, DOI 10.1090/S0002-9947-1948-0027750-7
- Erwin Kleinfeld, Generalization of alternative rings. I, II, J. Algebra 18 (1971), 304–325; ibid. 18 (1971), 326–339. MR 0274545, DOI 10.1016/0021-8693(71)90063-9
Bibliographic Information
- © Copyright 1973 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 39 (1973), 242-246
- MSC: Primary 17D05
- DOI: https://doi.org/10.1090/S0002-9939-1973-0313348-X
- MathSciNet review: 0313348