卷 213, 编号 2 (2016)

It is proved that a commutative ring with identity R is arithmetic (i.e., the ideal lattice of R is distributive) if and only if for any finitely generated (or any finitely presented) R-module M and any ideal I of R the equality I +AnnM = Ann(M/IM) holds.

Abstract. In this work, we study the question of sufficient conditions for strong indecomposability of Dedekind rings as finite rank modules over their Dedekind subrings. This question was considered earlier in the special case of rings of algebraic integers. Necessary and sufficient conditions are given for Galois extensions. The question of k-indecomposability is also considered.

Abstract. Algebraic simplification is considered from the general point of view. The main notion here is the so-called scheme of simplification due to the author. For a corepresented algebra, simplificators are related to the standard basis (Gröbner basis, Gröbner–Shirshov basis) of the ideal of defining relations. We introduce as a main subject for study the class of algebras that may be obtained from ordered semigroup algebras by “deformation” in some sense. This class contains free associative algebras and universal enveloping algebras of Lie algebras. Primary attention is paid to the latter algebras. Some possible applications to cryptography are given.

Abstract. In this paper, we give a consistent presentation of dimension properties and properties of bases for vector spaces over distributive lattices. The bases consisting of join irreducible vectors are studied and their uniqueness is proved. Criteria for the following are proved: the set of join irreducible vectors is a generating set for a vector space; this set is a vector space basis; all bases contain the same number of vectors. A criterion of uniqueness for the basis is proved. The basis containing the greatest number of vectors is found. We give a description for all standard bases of a vector space. We prove a theorem allowing one to calculate the space dimension and to find the basis of the smallest number of vectors by known algorithms. These results are applied to vector spaces over chains: we prove that there exists a standard basis, that the basis of join irreducible vectors is the standard basis, that a standard basis is unique. We calculate the dimension of the arithmetic space and describe all bases containing the smallest number of vectors. It is proved that all such bases are standard.

In this paper, we give examples of infinite series of finite rings B_v^(m) , where m ≥ 2, 0 ≤ v ≤ p−1, and p is a prime number, that are not representable by matrix rings over commutative rings, and we describe the basis of polynomial identities of these rings. We prove here that every variety var B_v^(m) , where m = 2 or m − 1 = (p − 1)k, k ≥ 1, and p ≥ 3 or p = 2 and m ≥ 3, 0 ≤ v < p, and p is a prime number, is a minimal variety containing a finite ring that is nonrepresentable by a matrix ring over a commutative ring. Therefore, we describe almost finitely representable varieties of rings whose generating ring contains an idempotent element of additive order p.

In this paper, the investigation of groups with additional finiteness conditions is continued. Criteria of residual finiteness and of almost local solvability of a periodic F* -group are obtained.

Let G be a topological group. For a function f : G → ℝ and h ∈ G, the right difference function Δ_hf is defined by Δ_hf(g) = f(gh) − f(g) (g ∈ G). A function H: G → ℝ is said to be additive if it satisfies the Cauchy functional equation H(g + h) = H(g) + H(h) for every g, h ∈ G. A class F of real-valued functions defined on G is said to have the difference property if, for every function f : G → ℝ satisfying Δ_hf ∈ F for every h ∈ G, there is an additive function H such that f − H ∈ F. The Erdős conjecture claiming that the class of continuous functions on ℝ has the difference property was proved by de Bruijn; later on, Carroll and Koehl proved a similar result for the compact Abelian groups and, under an additional assumption, for the compact metric groups, namely, under the assumption that all functions of the form ∇_hf(g) = f(hg)−f(g), g ∈ G, are Haar measurable for every h ∈ G. One of the consequences of this assumption is the boundedness of the function {g, h} ⟼ f(gh) − f(g) − f(h) + f(e), g, h ∈ G, for every function f on a compact group G for which the difference functions Δ_hf are continuous for every h ∈ G and the functions ∇_hf are Haar measurable for every h ∈ G (e stands for the identity element of the group G). In the present paper, we consider the difference property under the very strong assumption that the function {g, h} ⟼ f(gh) − f(g) − f(h) + f(e), g, h ∈ G, is bounded. This assumption enables us to obtain results concerning difference properties not only for functions on groups but also for functions on homogeneous spaces.