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Vol 237, No 2 (2019)
- Year: 2019
- Articles: 13
- URL: https://journal-vniispk.ru/1072-3374/issue/view/14987
Article
Primitive and Almost Primitive Elements of Schreier Varieties
Abstract
A variety of linear algebras is said to be Schreier if any subalgebra of a free algebra of this variety is free. A system of elements of a free algebra is primitive if there is a complement of this system with respect to a free generating set of the free algebra. An element of a free algebra of a Schreier variety is said to be almost primitive if it is not primitive in the free algebra, but it is a primitive element of any subalgebra that contains it. This survey article is devoted to the study of primitive and almost primitive elements of Schreier varieties.
157-179
Elementary Equivalence of Endomorphism Monoids of Almost Free S-Acts
Abstract
In this paper, we study the connection between elementary equivalence of endomorphism monoids of almost free S-acts (acts that are unions of projective indecomposable cyclic S-acts) and equivalence (in first- or second-order logic) of original monoids.
180-190
191-244
U-Projectors and Fields of U-Invariants
Abstract
We present a general construction of the U-projector (the homomorphism of an algebra into its field of U-invariants identical on the subalgebra of U-invariants). It is shown how to apply the U-projector to find the systems of free generators of the fields of U-invariants for representations of reductive groups.
245-253
Rolling Simplexes and Their Commensurability IV. A Farewell to Arms!*
Abstract
This text by pure algebraic reasons outlines why the spectrum of maximal ideals SpecℂA of a countable-dimensional differential ℂ-algebra A of transcendence degree 1 without zero divisors is locally analytic, which means that for any ℂ-homomorphism ψM : A → ℂ(M ∈ SpecℂA) and any a ∈ A the Taylor series \( {\overset{\sim }{\psi}}_M(a)\overset{\mathrm{def}}{=}\sum \limits_{m=0}^{\infty }{\psi}_M\left({a}^{(m)}\right)\frac{z^m}{m!} \) has nonzero radius of convergence depending on the element a ∈ A.
254-262
The Kostrikin Radical and Similar Radicals of Lie Algebras
Abstract
The existing notion of the Kostrikin radical as a radical in the Kurosh–Amitsur sense on classes of Mal’tsev algebras over rings with 1/6 is not completely justified. More precisely, to the fullest extent it is true for classes of Lie algebras over fields of characteristic zero and, as shown in the given paper, classes of algebraic Lie algebras of degree not greater than n over rings with 1/n! at all n ≥ 1. Similar conclusions are obtained in the paper also for the Jordan, regular, and extremal radicals constructed analogously to the Kostrikin radical.
263-279
280-283
Goldie Rings Graded by a Group with Periodic Quotient Group Modulo the Center
Abstract
In this paper, we study gr-prime and gr-semiprime Goldie rings graded by a group with periodic quotient group modulo the center. We enhance the theorem of Goodearl and Stafford (2000) about gr-prime rings graded by Abelian groups; we extend the Abelian group class to the class of groups with periodic quotient group modulo the center. We also decompose the orthogonal graded completion Ogr(R) of a gr-semiprime Goldie ring R (graded by a group satisfying the same condition) into a direct sum of gr-prime Goldie rings R1, . . . , Rn and prove that the maximal graded quotient ring Qgr(R) equals the direct sum of classical graded quotients rings of R1, . . . , Rn.
284-286
Uniqueness of Addition in Lie Algebras of Chevalley Type over Rings with 1/2 and 1/3
Abstract
In this paper, it is proved that Lie algebras of Chevalley type (An, Bn, Cn, Dn, E6, E7, E8, F4, and G2) over associative commutative rings with 1/2 (with 1/2 and 1/3 in the case of G2) have unique addition. As a corollary of this theorem, we note the uniqueness of addition in semisimple Lie algebras of Chevalley type over fields of characteristic ≠ 2 (≠ 2, 3 in the case of G2).
287-303
Numerical Characteristics of Varieties of Poisson Algebras
Abstract
This paper is a survey of recent results of investigations on varieties of Poisson algebras. We give constructions of varieties of Poisson algebras with extremal properties, we give equivalent conditions for the polynomial codimension growth of a variety of Poisson algebras, we study varieties of Poisson algebras whose ideals of identities contain the identity {x, y} ⋅ {z, t} = 0, and we study the interrelation between such varieties and varieties of Lie algebras.
304-322
The Intersection of the Powers of the Topological Jacobson Radical and Topological Krull Dimension
Abstract
In this paper, it is proved that a certain power of the topological Jacobson radical for a ring annihilates a left module having topological Krull dimension over this ring. The estimation of this power depends on the topological Krull dimension and the dual topological Krull dimension. A similar estimation for discrete Jacobson radical holds true. Levitzky’s theorem is generalized for topological rings.
323-328
329-331
332-335