Embedding Theorems for Abelian Groups with Valuations

Abstract

Introduetion. This study arose from an attempt to simplify the proof and deepen the eontent of Hahn's Embedding Theorem for ordered abelian groups (Hahn [8]). When doing this, it was discovered that the proper framework for such a diseussion is provided by assigning to every element the class of all commensurable elements as its value, and considering the structure of the groups in terms of the resulting valuation. An extension of this concept leads to the following general definition of a valuation of an abelian group. If A is an abelian operator group and r is a partially ordered set, then a Y-valuation of A is obtained by assigning to each a =7^= 0 in A a non-empty trivially ordered subset of r, called the set of values of a, subject to the requirement that if none of the values of a and b is greater than y (greater than or equal to y), then none of the values of a ? b and ra is greater than y (greater than or equal to y). An abelian operator group with a definite T-valuation is called a T-group. An example of a P-group is given by the following direct generalization of the groups introduced by Hahn. If, to each y in P, there is assigned an abelian operator group B(y) (with respect to a common operator domain R), then the Y-sum V of the B(y) is defined as follows: V is the totality of vectors b = (? ? ?,by, ? ? ?), with by in B(y) and by = 0 for all y with the exception of a set which satisfies the ascending chain condition. Addition and multiplication (by R) are defined componentwise. The values of b are the maximal y^s with by =^= 0. The subgroup G of V is a c-subgroup if for every y in r and b =? 0 in B(y), there exists an element c in 0 with value y such that cy = b. ois a T-isomorpMsm if oand 0-1 are isomorphisms that preserve values. In section 3 we prove the following embedding theorem: