ON SUBDIRECT UNIONS, I By L. FUCHS (Budapest) (Presented by G. HAJ6S) w 1. Introduction During the last decade the notion of subdirect union' has proved to be one of the most important and fruitful notions in abstract algebra. Indeed, a great number of significant problems from quite different fields of abstract algebra have found their solutions in terms of the notion of subdirect union. It turned out that in several cases without any hope of getting structure theorems by direct decompositions, subdil~ect unions may be used with great success. For example, let us mention N. JACOBSON'S structure theorem e generalizing the classical Wedderburn--Artin structure theorems from rings with minimal condition and no nilpotent ideals different from 0 to arbitrary rings with zero Jacobson radical, or the well-known result in the theory of infinite Abelian groups, according to which a torsion free Abelian group of finite rank n is a subdirect sum of n groups of rank one, each of which is isomorphic to some subgroup of the additive group of the rational numbers, 3 etc. The power of this concept may be judged even from the fact, proved by G. BIRKHOFF, 4 which states that any algebraic structure 5 may be repre- sented as a subdirect union of subdirectly irreducible algebraic structures. However, there is a great difference between the direct decompositions on the one hand, and the representations as subdirect unions on the other hand. If an algebraic structure G can be represented as the direct union of its substructures: 6 G-- A1 ~- A2 @ ..- -~ A,, then the Structure of G may be 1 For the definition see w 3 below. 2 JACOBSON [8]; cf. McCoy [9], and BeowN and McCoy [5]. (The numbers in brackets refer to the Bibliography given at the end of the paper.) a See e.g. BAEe [1]. 4 See B1RKHOFF [2], and also [3], pp. 91--92. 5 By an algebraic structure we mean an arbitrary system with binary operations satisfying certain laws and possibly having operator domains. (Cf. BOURBAKfs "structure alg~brique".) 6 For the sake of simplicity we restrict ourselves only to the case when the number of the components in the direct union is finite. We denote direct unions by the sign z, while @ is reserved for,-subdirect unions.