Volume 233, Nº 5 (2018)
- Ano: 2018
- Artigos: 22
- URL: https://journals.rcsi.science/1072-3374/issue/view/14948
Article
Congratulations
On Linear Groups with the Property of Order Finiteness of All Primitive Words in Generators
Resumo
It is well known that a finitely generated linear group of finite exponent is finite. It is proved in this paper that there exist infinite finitely generated linear groups such that all primitive words from generators have finite order.
On p-Adic Approximation of Sums of Binomial Coefficients
Resumo
We propose higher-order generalizations of Jacobsthal’s p-adic approximation for binomial coefficients. Our results imply explicit formulas for linear combinations of binomial coefficients \( \left(\underset{p}{ip}\right)\left(i=1,2,\dots \right) \)
Homomorphisms of Lie Groups
Resumo
In this paper, the authors describe homomorphisms of Lie groups into the groups u(R) of invertible elements of rings R for a large class of rings R, which contains, in particular, subrings of matrix rings and also Noetherian rings.
On the Additive Structure and Asymptotics of Codimensions cn in the Algebra F(5)
Resumo
In this paper, we investigate the additive structure of the algebra F(5), i.e., a relatively free, associative, countably-generated algebra with the identity [x1, . . . , x5] = 0 over an infinite field of characteristic ≠2, 3. We study the space of proper multilinear polynomials in this algebra and means of basis construction in one of its basic subspaces. As an additional result, we obtain estimations of codimensions cn = dimPn/Pn∩ T(5), where Pn is the space of multilinear polynomials of degree n in F(5) and T(5) is the T-ideal generated by the long commutator [x1, . . . , x5].
Serial Group Rings of Finite Simple Groups of Lie Type
Resumo
Suppose that F is a field whose characteristic p divides the order of a finite group G. It is shown that if G is one of the groups 3D4(q), E6(q), 2E6(q), E7(q), E8(q), F4(q), 2F4(q), or 2G2(q), then the group ring FG is not serial. If G = G2(q2), then the ring FG is serial if and only if either p > 2 divides q2− 1, or p = 7 divides \( {q}^2+\sqrt{3q}+1 \) but 49 does not divide this number.
Finite Combinatorial Generation of Metabelian T-Ideal
Resumo
In this work, we develop our idea on the construction of a system of combinatorial generators in a T-ideal of a free associative algebra, which is a full analogy of a Gröbner–Shirshov basis in a polynomial ideal. We prove a theorem on multilinear monomials that enables us to establish the existence of a finite set of combinatorial generators in a metabelian T-ideal.
On the Varieties of Commutative Metabelian Algebras
Resumo
The paper presents new results on varieties of commutative metabelian algebras over a field of zero characteristic. We study the structure of the multilinear part of the variety of all commutative metabelian algebras as a module of the symmetric group. Two almost nilpotent varieties are introduced and studied in this class of algebras. We prove the nonexistence of other almost nilpotent commutative metabelian varieties of subexponential growth.
The Atomic Theory of Division and Intersection of Semiring Ideals
Resumo
We consider two-sided ideals of semirings. More precisely, we study the theory of two-sided ideals in the signature consisting of the predicate symbol ⊆ and three function symbols that denote the intersection, right division, and left division of ideals. We prove the decidability of the set of those atomic formulas in this signature that are valid for all semirings and all valuations.
Once More on the Lattice of Subvarieties of the Wreath Product of the Variety of Semilattices and the Variety of Semigroups with Zero Multiplication
Resumo
It is known that the monoid wreath product of any two semigroup varieties that are atoms in the lattice of all semigroup varieties may have a finite as well as an infinite lattice of subvarieties. If this lattice is finite, then as a rule it has at most eleven elements. This was proved in a paper of the author in 2007. The exclusion is the monoid wreath product Sl w N2 of the variety of semilattices and the variety of semigroups with zero multiplication. The number of elements of the lattice L(Sl w N2) of subvarieties of Sl w N2 is still unknown. In a previous paper, we have shown that the lattice L(Sl w N2) contains a sublattice having 33 elements. In the present paper, it is proved that the lattice under consideration has exactly three maximal subvarieties. As a first application of the obtained results we calculate the finite basis of the lattice union of the variety of all semilattices and the largest variety among subvarieties of our lattice having at least one heterotypic identity. As a second application we show that the considered lattice of subvarieties has at least 39 elements
Group Ring Ideals Related to Reed–Muller Codes
Resumo
Reed–Muller codes are one of the most well-studied families of codes; however, there are till open problems regarding their structure. Recently a new ring-theoretic approach has emerged that provides a rather intuitive construction of these codes. This approach is centered around the notion of basic Reed–Muller codes. It is known that basic Reed–Muller codes ℳπ(m, k) over a prime field are powers of the radical RS of a corresponding group algebra and over a nonprime field there are no such equalities, except for trivial ones. In this paper, we consider the ideals ℜSℳπ(m, k) that arise while studying the inclusions of the basic codes and radical powers.
On the UA-Properties of Abelian sp-Groups and Their Endomorphism Rings
Resumo
An R-module A is said to be a UA-module if it is not possible to change the addition of A without changing the action of R on A. A semigroup (R, ·) is said to be a UA-ring if there exists a unique binary operation + making (R, ·, +) into a ring. In this paper, the UA-properties of sp-groups and their endomorphism rings are studied.
On Partially Ordered Rings
Resumo
A new notion of a partial ordering for rings is considered. Properties of arbitrary partially right \( \mathcal{K} \)-ordered rings are investigated. A series of results for linearly right \( \mathcal{K} \)-ordered rings is obtained. Some theorems are proved for ideals of those rings.
Specific Properties of One-Dimensional Pseudorepresentations of Groups
Resumo
We obtain assertions concerning general properties of one-dimensional (not necessarily bounded) pseudorepresentations of groups. In particular, we obtain a quantitative condition on the numerical defect of a given pseudorepresentation which is sufficient for the pseudorepresentation to be pure, i.e., for the restriction of the given pseudorepresentation to every amenable subgroup be an ordinary character of this subgroup.