Necessary conditions for the coperiodicity of quotients of direct products of Abelian groups

Abstract

The problem of describing the structure of the group P---Xli~zAi/~iAi has attracted the attention of specialists in the theory of Abelian groups. The task of investigating the structure of such a group was proposed in [I] (see Problem 24). It was proved in [2, 3] that the group P is algebraically compact whenever the set of indices is countable. This result had a considerable influence on the development of the theory of narrow groups, groups of extensions, radical classes, etc. The above-mentioned result was generalized in [4] to the case of K-direct sums. The search for necessary conditions for the group P to be algebraically compact was begun in [5, 6], and such a necessary condition was found there for quotients of a power of a single group. Finally, a necessary condition was found in [7] for algebraical compactness of an arbitrary group ~eIAi/H. It follows from [2, 3] that the for H=~,i~A~ , this condition is also sufficient. In view of the special importance of the class of coperiodic groups, the question of the coperiodicity of P is a very natural one. The main result of our article is Theorem i, which gives a necessary condition for the group OA~/H to be coperiodic. This result immediately implies a necessary condition for K the coperiodicity of I],=xAi/H (see Theorem 2). If K is a o-ideal of the Boolean algebra B (f), H = @Ai, where K I C K, then our necessary condition turns out to be sufficient as KI well (see Theorem 3). In Theorem 4, we give a necessary and sufficient condition for P to be coperiodic. In Theorem 5, we prove that there exists a maximal ideal KCB (I), such that the group @Ai/@A i is coperiodic. K KI All the groups considered below are assumed to be Abelian. The following notation will be adhered to throughout the article: i) B(I) is the Boolean algebra of all subsets of the set I.