Rings with idempotents in their nuclei

Rich, Michael

doi:10.1090/S0002-9947-1975-0371972-9

Rings with idempotents in their nuclei
HTML articles powered by AMS MathViewer

by Michael Rich PDF

Trans. Amer. Math. Soc. 208 (1975), 81-90 Request permission

Abstract:

Let $R$ be a prime nonassociative ring. If the set of idempotents of $R$ is a subset of the nucleus of $R$ or of the alternative nucleus of $R$ then it is shown that $R$ is respectively an associative or an alternative ring. Also if $R$ has one idempotent $\ne 0,1$ which is in the Jordan nucleus or in the noncommutative Jordan nucleus then it is shown that $R$ is respectively a Jordan or a noncommutative Jordan ring.

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
A. A. Albert, A theory of power-associative commutative algebras, Trans. Amer. Math. Soc. 69 (1950), 503–527. MR 38959, DOI 10.1090/S0002-9947-1950-0038959-X
Nathan Jacobson, Structure of rings, Revised edition, American Mathematical Society Colloquium Publications, Vol. 37, American Mathematical Society, Providence, R.I., 1964. MR 0222106
Nathan Jacobson, Structure and representations of Jordan algebras, American Mathematical Society Colloquium Publications, Vol. XXXIX, American Mathematical Society, Providence, R.I., 1968. MR 0251099, DOI 10.1090/coll/039
Kevin McCrimmon, Structure and representations of noncommutative Jordan algebras, Trans. Amer. Math. Soc. 121 (1966), 187–199. MR 188261, DOI 10.1090/S0002-9947-1966-0188261-2
J. Marshall Osborn, Commutative algebras satisfying an identity of degree four, Proc. Amer. Math. Soc. 16 (1965), 1114–1120. MR 184980, DOI 10.1090/S0002-9939-1965-0184980-7
J. Marshall Osborn, Varieties of algebras, Advances in Math. 8 (1972), 163–369 (1972). MR 289587, DOI 10.1016/0001-8708(72)90003-5
R. D. Schafer, Noncommutative Jordan algebras of characteristic $0$, Proc. Amer. Math. Soc. 6 (1955), 472–475. MR 70627, DOI 10.1090/S0002-9939-1955-0070627-0
Richard D. Schafer, An introduction to nonassociative algebras, Pure and Applied Mathematics, Vol. 22, Academic Press, New York-London, 1966. MR 0210757
Armin Thedy, On rings with completely alternative commutators, Amer. J. Math. 93 (1971), 42–51. MR 274537, DOI 10.2307/2373446