Simple (-1, 1) Rings with an Idempotent

Maneri, Carl

doi:10.2307/2033969

Cited by 7 publications

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“…If F is a simple, Lie admissible GRA ring with idempotent e^0,# 1, by Theorem 2, F is a simple, Lie admissible, right alternative ring. Such rings are simple (-1,1) rings, and by [8], they are associative.…”

Section: Proofsmentioning

confidence: 99%

Generalization of Right Alternative Rings

Hentzel¹,

Cattaneo²

1975

Transactions of the American Mathematical Society

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We study nonassociative rings R satisfying the conditions (1) (ab, c, d) + (a, b, [c, d]) = a(b, c, d) + (a, c, d)b for all a, b, c, d e R, and (2) (x, x, x) = 0 for all x S. R. We furthermore assume weakly characteristic not 2 and weakly characteristic not 3. As both (1) and (2) are consequences of the right alternative law, our rings are generalizations of right alternative rings. We show that rings satisfying (1) and (2) which are simple and have an idempotent + 0, # 1, are right alternative rings. We show by example that (x, e, e) may be nonzero. In general, R = R' + (R, e, e) (additive direct sum) where R' is a subring and (R, e, e) is a nilpotent ideal which commutes elementwise with R. We examine R' under the added assumption of Lie admissibility: (3) (a, b, c) + (Jb, c, a) + (c, a, b) = 0 for all a, b, cEJi, We generate the Peirce decomposition. If R' has no trivial ideals contained in its center, the table for the multiplication of the summands is associative, and the nucleus of R' contains R'Xq + R'qX. Without the assumption on ideals, the table for the multiplication need not be associative; however, if the multiplication is defined in the most obvious way to force an associative table, the new ring will still satisfy (1), (2), (3). 1. Introduction. We study nonassociative rings which satisfy the generalized right alternative law:

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Section: Proofsmentioning

confidence: 99%

Generalization of Right Alternative Rings

Hentzel¹,

Cattaneo²

1975

Transactions of the American Mathematical Society

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“…On the other hand, it is quite clear from (6), (7), and (8) that b also belongs to Bb. Now let C be the ideal generated by b.…”

Section: Introductionmentioning

confidence: 99%

“…(2) (x, y, z) + (y, z, x) + (z, x, y) = 0 for all x, y, z E R-Maneri [7] proved that a simple ring of type ( -1, 1) with characteristic prime to 6 having an idempotent e^Q, 1 is associative. It is shown in this paper that when R is a (-1, 1) ring with no trivial ideals which has characteristic prime to 6, then if R contains an idempotent e^Q, 1, it has a Peirce decomposition relative to e. Further, the multiplicative relations between the submodules of the Peirce decomposition relative to containment are the same as those for an associative ring.…”

Section: Introductionmentioning

confidence: 99%

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Prime (-1,1) rings with idempotent

Sterling¹

1967

Proc. Amer. Math. Soc.

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“…In this proof we use Albert's result that semisimple finite dimensional right alternative algebras are alternative [2], and Wedderburn's structure theorem that semisimple associative algebras are complete direct sums of matrix rings over division rings [3]. We use Maneri's result that for a ( -1, 1) algebra of characteristic 9^2, 3, (A, A, A) is an ideal [5], and we use the results in my paper [4] in several places.…”

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confidence: 99%

(-1,1) algebras

Hentzel¹

1970

Proc. Amer. Math. Soc.

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In a nonassociative ring A, the symbol (a, 6, c) where a, b, c are elements of A is defined as (a, b, c) = (ab)c -a(bc). The symbol [a, b] where a, 6 are elements of A is defined as [a, b] =ab -ba. The nucleus of A, N(A)= {nEA\ (re, a, 6) = (a, n, 6) = (a, 6, n) = 0 for all a, bEA ). The center of A, C(A) = {sEN(A)\ [s, a] =0 for all aEA). A trivial ideal of A is an ideal ly±0 of A such that P = 0.A ( -1, 1) ring A is a nonassociative ring in which the following identities are assumed to hold.(1) (a, b, c) + (a, c, b) = 0,for all a, 6, c elements of A. A ( -1, 1) algebra is a ( -1, 1) ring with identity which is also a finite dimensional vector space over a field F which satisfies a(ab) =a(ab) = (aa)b for all a, 6 elements of A, a in F. We shall not require that a subalgebra of A contain the identity of A, though we do require that it contain an identity of its own. When hypotheses are placed on a ring as a whole, often these hypotheses imply certain properties for the center. For example, the center of a simple ring is a field. It is possible then, that hypotheses placed on the center will be reflected in the structure of the whole ring. What hypotheses are possible? Keeping in mind that the center of a simple ring is a field, one might suggest we assume that the center is simple, or semisimple. We could assume there are no ideals of the whole ring contained in the center. Or, we might assume that the center has no nilpotent elements. In this paper we show that any one of these conditions is sufficient to make a ( -1, 1) algebra associative. We prove the theorem:Theorem.If A is a ( -1, 1) algebra over afield of characteristic 9^2, 3, and the center of A contains no trivial A ideals, then A is associative.In this proof we use Albert's result that semisimple finite dimensional right alternative algebras are alternative [2], and Wedderburn's structure theorem that semisimple associative algebras are complete direct sums of matrix rings over division rings [3]. We use Maneri's result that for a ( -1, 1) algebra of characteristic 9^2, 3, (A, A, A) is an ideal [5], and we use the results in my paper [4] in several places.

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Simple (-1, 1) Rings with an Idempotent

Cited by 7 publications

References 0 publications

Generalization of Right Alternative Rings

Generalization of Right Alternative Rings

Prime (-1,1) rings with idempotent

(-1,1) algebras

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