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On prime right alternative rings with commutators in the left nucleus

Published online by Cambridge University Press:  17 April 2009

Erwin Kleinfeld  and
Harry F. Smith
Erwin Kleinfeld
Affiliation:
Division of Mathematical SciencesUniversity of Iowa Iowa City, IA 52242United States of America
Harry F. Smith
Affiliation:
Department of Mathematics, Statistics and Computing ScienceUniversity of New England Armidale, NSW 2351, Australia
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Abstract

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A ring is called s–prime if the 2-sided annihilator of a nonzero ideal must be zero. In particular, any simple ring or prime (—1, 1) ring is s–prime. Also, a nonzero s–prime right alternative ring, with characteristic ≠ 2, cannot be right nilpotent. Let R be a right alternative ring with commutators in the left nucleus. Then R is associative in the following cases: (1) R is prime, with characteristic ≠ 2, and has an idempotent e ≠ 1 such that (e, e, R) = 0. (2) R is an algebra over a commutative-associative ring with 1/6, and R is either s–prime, or R is prime and locally (—1,1).

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 50 , Issue 2 , October 1994 , pp. 287 - 298
Copyright
Copyright © Australian Mathematical Society 1994

References

[1]Humm, M.M., ‘On a class of right alternative rings without nilpotent ideals’, J. Algebra 5 (1967), 164–174.CrossRefGoogle Scholar
[2]Kleinfeld, E., ‘Anticommutative elements in alternative rings’, J. Algebra 83 (1983), 65–71.CrossRefGoogle Scholar
[3]Kleinfeld, E. and Smith, H.F., ‘On simple rings with commutators in the left nucleus’, Comm. Algebra 19 (1991), 1593–1601.Google Scholar
[4]Miheev, I.M., ‘A certain identity in right alternative rings’, (in Russian), Algebra i Logika 8 (1969), 357–366.Google Scholar
[5]Miheev, I.M., ‘On prime right alternative rings’, (in Russian), Algebra i Logika 14 (1975), 56–60.Google Scholar
[6]Miheev, I.M., ‘On simple right alternative rings’, (in Russian), Algebra i Logika 16 (1977), 682–710.Google Scholar
[7]Nam, Ng Seong, ‘Right nucleus in right alternative algebras’, J. London Math. Soc. (2) 21 (1980), 456–464.CrossRefGoogle Scholar
[8]Paul, Y., ‘A note on (−1,1) rings’, Jn¯āābha 11 (1981), 107–109.Google Scholar
[9]Pchelincev, S.V., ‘Prime algebras and absolute divisors of zero’, (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), 79–100.Google Scholar
[10]Skornyakov, L.A., ‘Right alternative division rings’, (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 15 (1951), 177–184.Google Scholar
[11]Skosyrskiĭ, V.G., ‘Nilpotency in Jordan and right alternative algebras’, (in Russian), Algebra i Logika 18 (1979), 73–85.Google Scholar
[12]Skosyrskiĭ, V.G., ‘Right alternative algebras with minimality condition for right ideals’, (in Russian), Algebra i Logika 24 (1985), 205–210.Google Scholar
[13]Sterling, N.J., ‘Prime (−1,1) rings with idempotent’, Proc. Amer. Math. Soc. 18 (1967), 902–909.Google Scholar
[14]Thedy, A., ‘Right alternative rings’, J. Algebra 37 (1975), 1–43.CrossRefGoogle Scholar