A Wedderburn decomposition for certain generalized right alternative algebras

Smith, Harry F.

doi:10.1090/s0002-9939-1976-0419540-0

Cited by 2 publications

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“…Moreover, the property of local finiteness is a radical property in 91 if and only if for any algebra A of 91 any element a E A such that a 2 = 0 is an Engel element. It was found that any semiprime GILA-ring is right alternative [393], a finite-dimensional GRA-nil-algebra is solvable, a nil-semisimple finite-dimensional GRA-algebra is alternative, and in the class of generalized (-I, l)-algebras an analogue of the Wedderburn theorem on the break-off of the radical holds [636]. Alternative and Right Alternative Rings.…”

Section: Rings Ofmentioning

confidence: 99%

Nonassociative rings

Bakhturin¹,

Slinko²,

Shestakov³

1982

J Math Sci

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Results in the theory of nonassociative rings and related directions reviewed in RZhMatematika during the period 1972-1978 are discussed. Special attention is given to infinite-dimensional Lie algebras, Jordan algebras, alternative rings, Mal'tsev algebras, and varieties, representations, and radicals of nonassociative rings.Results in the theory of nonassociative rings and related directions reviewed in RZhMatematika during the period 1972-1978 are discussed. Special attention is given to infinite-dimensional Lie algebras, Jordan algebras, alternative rings, Mal'tsev algebras, and varieties, representations, and radicals of nonassociative rings. I. Infinite-DimensionalLie Algebras I. Books. In the Russian language there appeared during this period the successive parts of the book Lie Algebras of the series of Bourbaki [26, 27], the book of Dixmier [55], and the book of Kaplansky [91]. Two books are devoted to the theory of infinite-dimensional Lie algebras: Amayo and Stewart [240] and Bakhturin [257]. The books [365, 400, 716, 718] are written in a more traditional style. Universal enveloping algebras are also discussed in [284].2. Varieties of Lie Algebras. Although the first works properly devoted to varieties of Lie algebras appeared only in 1967 (V. A. Parfenov) and in 1968 (Yu. A. Bakhturin), the machinery and first major results of this theory actually appeared earlier in the works of A. I. Mal'tsev, A. L Kostrikin, and A. I. Shirshov. Moreover, in the hands of these specialists in this theory the theories of varieties of groups and associative algebras were advanced. The latter circumstance led to the situation that at first analogues of results in these theories were sought in varieties of Lie algebras. Thus, theorems were proved on the freeness of a semigroup of varieties of Lie algebras over a field of characteristic 0 [148], on the description of Sehreier varieties over such a field [ii], and a monotonicity theorem for verbal ideals [ii, 146, 49]. It was noted in [14] that the theorems on a semigroup of varieties and Sehreier varieties are also true in the case of an arbitrary infinitedimensional field; it is also shown there that in the case of a field of two elements the multiplication of varieties of Lie algebras is nonassociative. A description of Schreier varieties of Lie algebras, i.e., varieties in which any subalgebra of a free algebra is free, was given by M. V. Zaitsev for the case of any commutative ring: it is shown there that the Sehreier varieties are trivial; e.g., if the base ring is a field these are the varieties D, ~i, @ in standard notation.The problem of a finite basis was solved negatively in [705]. This example, which pertains to fields of characteristic 2, was later extended to the case of any prime characteristic p [67,68]. In [67] an example is also given of a (2p + 3)-dimensional Lie algebra over an infinite field of characteristic p > 0 without a finite basis of identities (in the case p = 2 there is also such an example in [705]). In [707] an example was constructed of an ...

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Section: Rings Ofmentioning

confidence: 99%