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Right alternative algebras with commutators in a nucleus
Published online by Cambridge University Press: 17 April 2009
Abstract
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Let A be a right alternative algebra, and [A, A] be the linear span of all commutators in A. If [A, A] is contained in the left nucleus of A, then left nilpotence implies nilpotence. If [A, A] is contained in the right nucleus, then over a commutative-associative ring with 1/2, right nilpotence implies nilpotence. If [A, A] is contained in the alternative nucleus, then the following structure results hold: (1) If A is prime with characteristic ≠ 2, then A is either alternative or strongly (–1, 1). (2) If A is a finite-dimensional nil algebra, over a field of characteristic ≠ 2, then A is nilpotent. (3) Let the algebra A be finite-dimensional over a field of characteristic ≠ 2, 3. If A/K is separable, where K is the nil radical of A, then A has a Wedderburn decomposition
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- Copyright © Australian Mathematical Society 1992
References
[1]Albert, A.A., ‘The structure of right alternative algebras’, Ann. of Math. 59 (1954), 408–417.CrossRefGoogle Scholar
[2]Dorofeev, G.V., ‘The nilpotency of right alternative rings’, (Russian), Algebra i Logika 9 (1970), 302–305.Google Scholar
[4]Hentzel, I.R., ‘Nil semi-simple (–1, 1) rings’, J. Algebra 22 (1972), 442–450.CrossRefGoogle Scholar
[5]Humm, M.M., ‘On a class of right alternative rings without nilpotent ideals’, J. Algebra 5 (1967), 164–174.CrossRefGoogle Scholar
[6]Kleinfeld, E., ‘On a class of right alternative rings’, Math. Z. 87 (1965), 12–16.CrossRefGoogle Scholar
[7]Kleinfeld, E. and Smith, H.F., ‘On simple rings with commutators in the left nucleus’, Comm. Algebra 19 (1991), 1593–1601.Google Scholar
[8]Miheev, I.M., ‘The theorem of Wedderburn on the splitting of the radical for a (–1, 1) algebra’, (Russian), Algebra i Logika 12 (1973), 298–304.Google Scholar
[9]Nam, Ng Seong, ‘Alternative nucleus of right alternative algebras’, Southeast Asian Bull. Math. 10 (1986), 149–154.Google Scholar
[10]Pchelincev, S.V., ‘Nilpotency of the associator ideal of a free finitely generated (–1, 1) ring’, (Russian), Algebra i Logika 14 (1975), 543–572.Google Scholar
[11]Pchelincev, S.V., ‘The locally nilpotent radical in certain classes of right alternative rings’, (Russian), Sibirsk. Mat. Zh. 17 (1976), 340–360.Google Scholar
[12]Roomel'di, R.E., ‘Nilpotency of ideals in a (–1, 1) ring with minimum condition’, (Russian), Algebra i Logiki 12 (1973), 333–348.Google Scholar
[13]Schafer, R.D., ‘The Wedderburn principal theorem for alternative algebras’, Bull. Amer. Math. Soc. 55 (1949), 604–614.CrossRefGoogle Scholar
[14]Skosyrskiī, V.G., ‘Right alternative algebras’, (Russian), Algebra i Logika 23 (1984), 185–192.Google Scholar
[15]Skosyrskiī, V.G., ‘Right alternative algebras with minimality condition for right ideals’, (Russian), Algebra i Logika 24 (1985), 205–210.Google Scholar
[16]Slin'ko, A.M., ‘The equivalence of certain nilpotencies of right alternative rings’, (Russian), Algebra i Logika 9 (1970), 342–348.Google Scholar
[17]Smith, H.F., ‘Finite-dimensional locally (–1, 1) algebras’, Comm. Algebra 7 (1979), 177–191.CrossRefGoogle Scholar
[18]Sterling, N.J., ‘Prime (–1, 1) rings with idempotent’, Proc. Amer. Math. Soc. 18 (1967), 902–909.Google Scholar
[20]Thedy, A., ‘Right alternative algebras and Wedderburn principal theorem’, Proc. Amer. Math. Soc. 72 (1978), 427–435.CrossRefGoogle Scholar
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