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We describe the ideals for Mizuhara extensions and find some necessary and sufficient conditions for the simplicity of the direct Mizuhara extension. Also, we study the Mizuhara construction for the matrix algebra and Burde algebras. We construct some various generalizations of the Mizuhara construction and exhibit some examples of the simple pre-Lie algebras that are obtained by this construction; in particular, we construct the simple Witt doubles and for a unital associative commutative algebra with derivation . [ABSTRACT FROM AUTHOR]

In 1974 Kharchenko proved that if a -component of an -graded associative algebra is PI then this algebra is PI. In the Novikov algebras of characteristic 0 the existence of a polynomial identity is equivalent to the solvability of the commutator ideal. We study a -graded Novikov algebra and prove that if the characteristic of the basic field is not 2 or 3 and its 0-component is associative or Lie-nilpotent of index 3 then the commutator ideal is solvable. [ABSTRACT FROM AUTHOR]

We describe the right-symmetric algebras of matrix type over a field of characteristic such that the left action of the orthogonal idempotents of is diagonalizable, and the right-module part includes no constant bichains. We construct some wide class of nonassociative algebras , where is a subalgebra and a right module over an associative algebra . We give a criterion for these algebras to be right-symmetric. Assuming that , we show that the algebras of this class are either simple or local. We exhibit some examples of simple right-symmetric algebras and right-symmetric algebras without nilpotent right ideals whose right-module part is not an irreducible module over . [ABSTRACT FROM AUTHOR]

Product theorems are available for associative algebras over a scalar ring containing as well as for such algebras of rank 3. We prove the exact analog of a product theorem for alternative algebras of rank 3 and describe the identities in three variables for the alternative algebras Lie nilpotent of a given degree. We also prove an analog of a product theorem in a general form without any restriction on the algebra rank and show that there is no exact analog of a product theorem for alternative algebras. The connection between the concepts of the Lie and strong Lie nilpotency is studied for a given algebra of rank 3 and its multiplication algebra. [ABSTRACT FROM AUTHOR]

We construct an additive basis for the relatively free associative algebra F(5)(K) with the Lie nilpotency identity of degree 5 over an infinite domain K containing 1 6 . We prove that approximately half of the elements in F(5)(K) are central. We also prove that the additive group of F(5)(ℤ) lacks the elements of simple degree ≥ 5. We find an asymptotic estimation of the codimension of T-ideal, which is generated by the commutator [x1,x2,...,x5] of degree 5. [ABSTRACT FROM AUTHOR]

We prove that the group of tame automorphisms of a free Lie algebra of rank 3 (as well as of a free anticommutative algebra) over an arbitrary integral domain has the structure of an amalgamated free product. We construct an example of a wild automorphism of a free Lie algebra of rank 3 (as well as of a free anticommutative algebra) over an arbitrary Euclidean ring analogous to the Anick automorphism [1] of free associative algebras. [ABSTRACT FROM AUTHOR]

We study almost commutative L-varieties of vector spaces. We describe nonnilpotent almost commutative L-varieties generated by an associative algebra, which is considered as a vector space. [ABSTRACT FROM AUTHOR]

We establish connection between the differentiably simple associative commutative algebras with unity and the simple Filippov algebras. [ABSTRACT FROM AUTHOR]

Given a finite-dimensional associative commutative algebra A over a field F, we define the structure of a Lie algebra using a nonzero derivation D of A. If A is a field and char F > 3; then the corresponding algebra is simple, presenting a nonisomorphic analog of the Zassenhaus algebra W ( m). [ABSTRACT FROM AUTHOR]

Some answer is obtained to the problem by A. R. Kemer: We show that for a given number k and all sufficiently large p every prime variety of associative algebras of matrix type k over a field of characteristic p > 0 is regular. [ABSTRACT FROM AUTHOR]

Consider the sequence c( V) of codimensions of a variety V of Poisson algebras. We show that the growth of every variety V of Poisson algebras over an arbitrary field is either bounded by a polynomial or at least exponential. Furthermore, if the growth of V is polynomial then there is a polynomial R( x) with rational coefficients such that c( V) = R( n) for all sufficiently large n. We present lower and upper bounds for the polynomials R( x) of an arbitrary fixed degree. We also show that the varieties of Poisson algebras of polynomial growth are finitely based in characteristic zero. [ABSTRACT FROM AUTHOR]

We describe the ternary derivations of finite-dimensional separable associative and Jordan algebras. [ABSTRACT FROM AUTHOR]

We study Leibniz algebras graded by finite root systems of type C. [ABSTRACT FROM AUTHOR]

Consider the algebra UT of upper triangular matrices of size s over an arbitrary field. Petrogradsky proved that the exponent of an arbitrary subvariety in var( UT) exists and is an integer. We strengthen the estimates for the growth of these varieties and provide equivalent conditions for finding these exponents. Kemer showed that in the case of a ground field of characteristic zero there exists no varieties of associative algebras with growth intermediate between polynomial and exponential. We prove that this property extends to the case of the fields of arbitrary characteristic distinct from 2. [ABSTRACT FROM AUTHOR]

We establish the composition-diamond lemma for associative nonunitary Rota-Baxter algebras of weight λ. To give an application, we construct a linear basis for a free commutative and nonunitary Rota-Baxter algebra, show that every countably generated Rota-Baxter algebra of weight 0 can be embedded into a two-generated Rota-Baxter algebra, and prove the 1-PBW theorems for dendriform dialgebras and trialgebras. [ABSTRACT FROM AUTHOR]

We prove that every prime variety of associative algebras over an infinite field of characteristic p>0 is generated by either a unital algebra or a nilalgebra of bounded index. We show that the Engel verbally prime T-ideals remain verbally prime as we impose the identity $$ x^{p^N } = 0 $$ for sufficiently large N. We then describe all prime varieties in an interesting class of varieties of associative algebras. [ABSTRACT FROM AUTHOR]

We prove that the Lie algebra of skew-symmetric elements of the free associative algebra of rank 2 with respect to the standard involution is generated as a module by the elements [ a, b] and [ a, b]3, where a and b are Jordan polynomials. Using this result we prove that the Lie algebra of Jordan derivations of the free Jordan algebra of rank 2 is generated as a characteristic F-module by two derivations. We show that the Jordan commutator s-identities follow from the Glennie-Shestakov s-identity. [ABSTRACT FROM AUTHOR]

We describe all homogeneous structures of Rota-Baxter algebras on a 0-dialgebra with associative bar-unity and give a corollary on the structure of a Rota-Baxter algebra on an arbitrary associative dialgebra with bar-unity as well as a unital associative conformal algebra. We prove that an arbitrary alternative dialgebra may be embedded into an alternative dialgebra with associative barunity. We suggest the definition of variety of dialgebras in the sense of Eilenberg which is equivalent to that introduced earlier by Kolesnikov. [ABSTRACT FROM AUTHOR]

We consider some algebraical systems that lead to various nearly associative triple systems. We deal with a class of algebras which contains Leibniz-Poisson algebras, dialgebras, conformal algebras, and some triple systems. We describe all homogeneous structures of ternary Leibniz algebras on a dialgebra. For this purpose, in particular, we use the Leibniz-Poisson structure on a dialgebra. We then find a corollary describing the structure of a Lie triple system on an arbitrary dialgebra, a conformal associative algebra and a classical associative triple system. We also describe all homogeneous structures of an ( ε, δ)-Freudenthal-Kantor triple system on a dialgebra. [ABSTRACT FROM AUTHOR]

We describe the simple special unital Jordan superalgebras with associative even part A whose odd part M is an associative module over A. We prove that each of these superalgebras, not isomorphic to a superalgebra of nondegenerate bilinear superform, is isomorphically embedded into a twisted Jordan superalgebra of vector type. We exhibit a new example of a simple special Jordan superalgebra. We also describe the superalgebras such that M ∩ [A,M] ≠ 0. [ABSTRACT FROM AUTHOR]

We prove that the groups of tame automorphisms of a free Lie algebra (free associative algebra, absolutely free algebra, algebra of polynomials) of rank at least four over a field of characteristic zero admit no faithful representation by matrices over any field. [ABSTRACT FROM AUTHOR]

We obtain a necessary and sufficient condition for a given collection of elements to freely generate a free associative algebra. We present some necessary conditions for primitivity of an element of a free associative algebra of rank 2. [ABSTRACT FROM AUTHOR]

We prove that each (possibly not finitely generated) associative PI-algebra does not coincide with its commutant. We thus solve I. V. L'vov's problem in the Dniester Notebook. The result follows from the fact (also established in this article) that, in every T-prime variety, some weak identity holds and there exists a central polynomial (the existence of a central polynomial was earlier proved by A. R. Kemer). Moreover, we prove stability of T-prime varieties (in the case of characteristic zero, this was done by S. V. Okhitin who used A. R. Kemer's classification of T-prime varieties). [ABSTRACT FROM AUTHOR]

Let k be a field of characteristic 0 and let k be a free associative algebra with finite basis X. Let $R=R(k,X)$ be the universal enveloping algebra of the square of Lie(X), regarded as a subalgebra of k and called the Specht subalgebra of the free algebra. We prove that k is a free (left) R-module, find sufficient conditions for some system of elements in k to be a basis for this module, and obtain an explicit formula that allows us to calculate the R-coefficients of the elements of the free algebra over a special basis of “symmetric monomials.” [ABSTRACT FROM AUTHOR]