Про застосування деяких понять теорiї кiлець для вивчення впливу систем пiдгруп групи

Abstract

Let A be a partially ordered set. For a, b ∈ A, we put [a, b] = {x ∈ A | a <= x <= b}. The deviation of A, denoted as dev(A), is defined by the following rule. If A is trivial, then we put dev(A) = −∞. If A is not trivial but satisfies the minimal condition, then dev(A) = 0. For a general ordinal , we define dev(A) = a provided dev(A) /= b and, in any descending chain a1 >= a2 >= · · · >= an > · · · of elements of A, all but finitely many of the closed intervals [an, an+1] have deviation less than a. Let G be a group and let S be some family of subgroups of G. Then S is partially ordered by inclusion. If a partially ordered set S has a deviation, then we will say that a family S has the Krull dimension. In this paper, we study the groups, in which the family Lnon-nn(G) of all non nearly normal subgroups has the Krull dimension. A subgroup H of the group G is said to be nearly normal, if H has finite index in its normal closure.