Prime (-1,1) rings with idempotent

Sterling, Nicholas J.

doi:10.1090/s0002-9939-1967-0215893-1

Cited by 3 publications

(1 citation statement)

References 11 publications

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“…Then A is associative under condition (i) by Corollary 2 to Theorem 3 in [19]. If A is prime, then A is associative under condition (ii) by Theorem 5 in [4], and under condition (iii) by Theorem 2 in [18]. Under condition (iv), the locally nilpotent radical of A is nilpotent by Theorem 3 in [12].…”

Section: Corollary Let a Be A Prime Right Alternative Algebra With Cmentioning

confidence: 99%

Right alternative algebras with commutators in a nucleus

Kleinfeld

Smith

1992

Bull. Austral. Math. Soc.

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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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Section: Corollary Let a Be A Prime Right Alternative Algebra With Cmentioning

confidence: 99%

Right alternative algebras with commutators in a nucleus

Kleinfeld

Smith

1992

Bull. Austral. Math. Soc.

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A generalization of strongly (−1, 1) rings

Kleinfeld

1988

Journal of Algebra

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(-1,1) rings

Hentzel¹

1969

Proc. Amer. Math. Soc.

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The center of a nonassociative ring is the set of all elements which commute and associate with all other elements of the ring, i.e. \x\ [x, a] = ix, a, b) = ia, x, b) = ix, a, b)=0 for all elements a, b}. Hypotheses on the ring imply conditions on the center, e.g. the center of any simple nonassociative ring is either 0 or a field. We examine the reverse problem. We place hypotheses on the center and see how they are reflected in properties of the ring.A ( -1, 1) ring is a nonassociative ring in which the following identities are assumed to hold:for all elements a, b, c. We study ( -1, 1) rings with an idempotent 5¿0, 1 with the further property that their centers contain no trivial ideals. (An ideal 7 is trivial if 7=^0 but 72 = 0.) Except for the cases of characteristic 2 and 3, we will be able to show that such an algebra has a Peirce decomposition A =An+Aio-\-Aoi+Aoo where the multiplication table for the summands is the same as if A were associative. We will also show that ^4ioj4oi+^4io+^4oi+^4oi^4io is an ideal of A contained in the nucleus of A. (The nucleus of a ring A is \x\ (a, b, x) = (a, x, b) = (a, b, x) =0 for all a, b EA }.) This latter statement was proved by Sterling [5 ] using the stronger hypothesis that the ring was free of trivial ideals entirely.

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Prime (-1,1) rings with idempotent

Cited by 3 publications

References 11 publications

Right alternative algebras with commutators in a nucleus

Right alternative algebras with commutators in a nucleus

A generalization of strongly (−1, 1) rings

(-1,1) rings

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