Titlebook: Selected Works of A.I. Shirshov; Leonid Bokut,Ivan Shestakov,Efim Zelmanov Book 2009 Birkh�user Basel 2009 Lie.Lie algebra.Shirshov, Anato

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On Free Lie RingsLet Σ be a commutative associative ring with unit, let .={..} be some set of symbols, and let . be the free associative algebra over Σ with free generating set . In the ring ., the set . generates a Lie subring . with respect to the operations . o ., addition, and scalar multiplication by elements of Σ.

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Some Algorithmic Problems for Lie AlgebrasIn a previous work [.] the author considered some algorithmic problems in the theory of ε-algebras. The same paper mentioned some literature relevant to these problems.

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TOM

On the Definition of the Binary-Lie PropertyIn the present note we construct an example of an algebra over a field of characteristic 2 that satisfies the identical relations . but is not binary-Lie. This example has been announced earlier [.]. For the necessary definitions and the history of the problem, see for example the work [.].

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Leonid Bokut,Ivan Shestakov,Efim ZelmanovEnglish translations of Shirshov’s works, which were published only in Russian.Due to Shirshov‘s future leading and unique handling of algebraic problems, the results are still of high interest for cu

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Altitude

Some Problems in the Theory of Rings that are Nearly Associativeion. Among other questions that remain outside the scope of the article, we mention, for example, various theorems about decomposition of algebras (see for example [., .]) which are closely related to the theory of algebras of finite dimension.

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聋子

On Some Identical Relations for Algebrasassociative algebras. It is shown there that, under natural assumptions on the characteristic, any such identical relation is a linear combination of the following relations: . where the summations in relations (1) and (2) are performed over all substitutions, and .. is the number of inversions in the permutation (.., .., ..) of 1, 2, 3.

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On Special ,-ringser, we will consider Jordan algebras. over an arbitrary ring of coefficients Σ, assuming only that Σ is a unital ring and that for each element . in the Jordan algebra there exists a unique element . such that 2.. Clearly, in this case the equation 2.=0 implies .=0. In such Jordan algebras, i.e., Jo