Generalizing alternative rings

Abstract. If / is the ideal generated by all associators, (a, b, c) = (ab)ca{bc), it is well known that in any nonassociative algebra R, I C (R, R, R) + R(R, R, R). We examine nonassociative algebras where / C (R, R, R). Such algebras include (-1, 1) algebras, Lie algebras, and, as we show, a large number of associator dependent algebras. An alternator is an associator of the type (a, a, b), {a, b, a), (f>, a, a). We next study algebras where the additive span of all alternators is an ideal. These include all algebras where I = {R, R, R) as well as alternative algebras. The last section deals with prime, right alternative, alternator ideal algebras satisfying an identity of the form [x, (x, x, a)] = y(x, x, [x, a]) for fixed y. With two exceptions, if this algebra has an idempotent e such that (e, e, R) = 0, then the algebra is alternative. All our work deals with algebras with an identity element over a field of characteristic prime to 6. All our containment relations are given by identities.

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“…The algebra (0, 0, 0, 0, 1) was given in [3] and [5, Equation (5) A (-1, 1) algebra is of the type (\, -\, 0, -{-, «-).…”

Section: Corollarymentioning

confidence: 99%

Alternator and associator ideal algebras

Hentzel¹,

Cattaneo²,

Floyd³

1977

Trans. Amer. Math. Soc.

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“…In their papers they assume the existence of an idempotent and prove that with the further assumption of simplicity [1] or primeness [4], then these rings are alternative.…”

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confidence: 99%

A Note on Generalizing Alternative Rings

Hentzel¹,

Cattaneo²

1976

Proceedings of the American Mathematical Society

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Abstract.Let R be a nonassociative ring of characteristic different from 2 and 3 which satisfies the following identities:(i) (ab, c, d) + (a, b, [c, d]for all a, b, c, d G R and with x ° y = (xy + yx)/2. We prove that if R is semiprime, then R is alternative.Introduction. In this paper we study rings R which satisfy the followingfor all a, b,c,d E R. We assume characteristic different from 2 and 3 and we define the ° product by x ° y = (xy + yx)/2. Identities (i), (ii) and (iii) hold in any alternative ring; thus these rings are generalizations of alternative rings.Rings of this kind have been studied by Getu and Rodabaugh. In their papers they assume the existence of an idempotent and prove that with the further assumption of simplicity [1] or primeness [4], then these rings are alternative.In our paper we do not require the existence of an idempotent, and we prove that any semiprime ring of characteristic different from 2 and 3 which satisfies (i), (ii) and (iii) is alternative.To prove this, we make use of the result that we stated in [2] that any semiprime ring of weakly characteristic different from 2 and 3 which satisfies (i) and (ii) is right alternative. It is known that in a right alternative ring the following identities hold (see [3]):

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“…Let A be a generalized alternative algebra over a field of characteristic ¥= 2 which has an idempotent e # 1, and suppose moreover that A has a Peirce decomposition relative to e. Throughout we shall use the convention that w«, x¡j,y¡j, z¡j e A¡j for i,j = 0, 1. In the subsequent lemma we shall use, often without specific mention, the multiplication for subspaces established in Theorem 4.1 of [2]; namely A¡¡Aj¡ C Ay, A ¡¡Ay C A« for i # j, and otherwise AmnAst C SnsAmt. We shall also use the License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use following facts established in Lemma 4.2, 4.3 and 5.2 of [2], where again /' ^ j:…”

mentioning

confidence: 99%

“…In the subsequent lemma we shall use, often without specific mention, the multiplication for subspaces established in Theorem 4.1 of [2]; namely A¡¡Aj¡ C Ay, A ¡¡Ay C A« for i # j, and otherwise AmnAst C SnsAmt. We shall also use the License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use following facts established in Lemma 4.2, 4.3 and 5.2 of [2], where again /' ^ j:…”

mentioning

confidence: 99%

A Wedderburn decomposition for certain generalized right alternative algebras

Smith¹

1976

Proc. Amer. Math. Soc.

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Generalizing alternative rings

Cited by 10 publications

References 8 publications

Alternator and associator ideal algebras

Alternator and associator ideal algebras

A Note on Generalizing Alternative Rings

A Wedderburn decomposition for certain generalized right alternative algebras

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