Vol 233, No 5 (2018)

This paper contains new and known results on formal matrix rings close to regular. The main results are given with proofs.

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 \( \left(\underset{p}{ip}\right)\left(i=1,2,\dots \right) \)

In this paper, we prove that for a linearly ordered skewfield the groups of quotients of the semigroup G_n(????) coincides with the group GL_n(????) for n ≥ 3.

We prove that B + AnnM = Ann(M/MB) for every finitely generated right module M over a strongly regular ring A and every ideal B of the ring A.

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 c_n = dimP_n/P_n∩ T⁽⁵⁾, where P_n 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].

We describe projective and injective acts over completely simple semigroups. Projective covers and injective hulls of acts over such semigroups are constructed.

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.

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.

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 ℳ_π(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 ℜ_Sℳ_π(m, k) that arise while studying the inclusions of the basic codes and radical powers.

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.

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.