Search

Given an affine algebraic variety X, we prove that if the neutral component \mathrm {Aut}^\circ (X) of the automorphism group consists of algebraic elements, then it is nested, i.e., is a direct limit of algebraic subgroups. This improves our earlier result (see Perepechko and Regeta [Transform. Groups 28 (2023), pp. 401–412]). To prove it, we obtain the following fact. If a connected ind-group G contains a closed connected nested ind-subgroup H\subset G, and for any g\in G some positive power of g belongs to H, then G=H. [ABSTRACT FROM AUTHOR]

In (Moutassim, n.d), we have proven that if A is an absolute valued algebra containing a nonzero central algebraic element, then A is a pre-Hilbert algebra. Here we show that A is finite dimensional in the following cases: 1) A satisfies (x², x, x) = 0 or (x, x, x²) = 0 2) A satisfies (x², x², x) = 0 or (x, x², x²) = 0. In these cases A is isomorphic to R, C, H or O. [ABSTRACT FROM AUTHOR]

Let A be an absolute valued algebra containing a nonzero central algebraic element. Then A is a pre-Hilbert algebra and is finite dimensional in the following cases: 1) A satisfies (x, x, x) = 0. 2) A satisfies (x², x²,x²) = 0. 3) A satisfies (x, x², x) = 0. In these cases A is isomorphic to R, C, H, O, C, H or O. It may be conjectured that every absolute valued algebra containing a nonzero central element is pre-Hilbert algebra. [ABSTRACT FROM AUTHOR]

We study the class of Bernstein algebras that are algebraic, in the sense that each element generates a finite-dimensional subalgebra. Every Bernstein algebra has a maximal algebraic ideal, and the quotient algebra is a zero-multiplication algebra. Several equivalent conditions for a Bernstein algebra to be algebraic are given. In particular, known characterizations of Bernstein train algebras in terms of nilpotency are generalized to the case of locally train algebras. Along the way, we show that if a Banach Bernstein algebra is algebraic (respectively, locally train), then it is of bounded degree (respectively, train). Then we investigate the Kurosh problem for Bernstein algebras: whether a finitely generated Bernstein algebra which is algebraic of bounded degree is finite-dimensional. This problem turns out to have a closed link with a question about associative algebras. In particular, when the bar-ideal is nil, the Kurosh problem asks whether a finitely generated Bernstein train algebra is finite-dimensional. We prove that the answer is positive for some specific cases and for low degrees, and construct counter-examples in the general case. On the other hand, by results of Yagzhev the Jacobian conjecture is equivalent to a certain statement about Engel and nilpotence identities of multioperator algebras. We show that the generalized Jacobian conjecture for quadratic mappings holds for Bernstein algebras. [ABSTRACT FROM AUTHOR]

Companions of the field of rational numbers and a real-closed algebraic expansion of the field of rational numbers are studied. The description of existentially closed companions of a real-closed algebraic expansion of a field of rational numbers refers to the field of study of classical algebraic structures. The general theory of companions and existentially closed companions, built on the basis of Fraisse's classes in the works of A.T. Nurtazin, is included in the classical field of existentially closed theories in model theory. The basic concept of a companion: two models of the same signature are called companions if for any finite submodel of one of them, there is an isomorphic finite submodel in the other. This approach, applied to specific classical structures and their theories, provides new tools for the study of these objects. The study of the companion class of rational and algebraic real number fields reveals companion fields containing transcendental and possibly algebraic elements with special properties of polynomials defining these elements. [ABSTRACT FROM AUTHOR]