The Double Chain Condition in Cyclic Operator Groups

Abstract

Naturally one wonders whether condition (b) is a consequence of (a) and added strength is given to such a conjecture, if one remembers Hopkins' 3 Theorem to the effect that in rings possessing an identity the ascending chain condition for right-ideals is a consequenice of the descending chain condition. However, it is possible to construct examples of cyclic groups where the descending, though not the ascenidinig, chain conditioni is satisfied by the admissible subgroups. In the light of Hopkins' Theorem, just quoted, it is only natural to assume that there will exist a large class of groups where cyclicity and descending chain condition imply the ascending chain condition; and it is the object of the present note to exhibit such classes of groups. Let A be an abelian group where the composition is written as addition a + b; and assume that A admits the elements in the system M as operators (= right multipliers). The M-subgroup S of A is said to be of length n = n(S), if the M-group S possesses a compositioni series 4 of length n. An ill-subgroup J of A is termed minimal, if Jt# 0 and if J does not