ABSTRACT
K. Benabdallah and M.A. Ouldbeddi [B096a] considered finite essential ex tensions of arbitrary torsion-free abelian groups as a natural generalization of al most completely decomposable groups. A group X is a fin ite essen tia l ex tension of a group A if there is an embedding φ : A —> X such that Χ /(Α φ) is finite and Αφ is essential in X . The latter means that every non-zero subgroup of X intersects Αφ non-trivially. If A is torsion-free, then its finite extension X is torsion-free if and only if Αφ is essential in X . Without loss of generality we may identify A with its isomorphic copy Αφ and assume that A is actually a subgroup of X . We are mainly interested in the case when A is completely decomposable of finite rank and X is almost completely decomposable. However, many points are obscured by displaying irrelevant structure of A. In this chapter we will restrict attention to cyclic essen tia l ex tensions , i.e., we consider torsion-free groups X that have a presentation
so that X /A is cyclic. Cyclic essential extensions are by no means trivial. They provide an abundant supply of examples for all sorts of phenomena. Later we will see that an arbitrary finite essential extension X of A can be described in the form
where k is a positive integer, Z* is the set of all integral 1 x k matrices (or row vectors), N is a k x k integral matrix with d e tN Φ 0, and a 1 is a k x 1 matrix (or column vector) with entries from A. Juxtaposition stands for the usual matrix multiplication. In order to deal with the general situation it is necessary to develop a theory of greatest common divisors of an integral matrix N and a column matrix a *■ of entries from A. These results first appeared in [B096a], [BM98a], and [BM98b].
