Date of Award
Doctor of Philosophy (PhD)
A number of years ago, I. Kaplansky raised informally the question of whether every valuation ring could be expressed as a homomorphic image of a valuation domain. By a valuation ring, we mean a commutative ring with identity whose ideals are linearly ordered by inclusion. The classical notion of a valuation ring included the assumption that the ring is a domain. For two cases an affirmative answer to Kaplansky's question is known: for 0-dimensional valuation rings; and, for valuation rings which are monoid rings. In the early 40's, Kaplansky obtained structure theorems for a large class of (maximally complete) valuation domains. We approached his question with the idea of attempting to generalize his techniques in order to obtain anologous structure theorems for valuation rings. The hope was that, as a by-product, these theorems would yield the required valuation domain and homomorphism. In this paper are the results and questions which were discovered in the search for these structure theorems. In Chapter 1, several definitions (e.g., immediate extension, maximal completion, pseudo-convergent sequence) and results are extended from valuation domains to valuation rings and, when appropriate, more generally to quasi-local rings. In particular, it is shown that a valuation ring which is maximal is also maximally complete. In Chapter 2, after generalizing the concept of value group to that of value monoid, we define a long power series ring for such a monoid, with coefficients in a field, and prove such a ring is a homomorphic image of a valuation domain. In Chapter 3, we obtain a bound on the cardinality of a valuation ring which depends only on the cardinalities of the residue field and value monoid, then show every valuation ring has a maximal completion. Kaplansky's work on maximally complete valuation domains (valued fields) is outlined in Chapter 4. A discussion of what had been hoped for and remarks on several unanswered questions, along with miscellaneous results, conclude Chapter 4.
Vicknair, J. Paul, "On Valuation Rings as Homomorphic Images of Valuation Domains." (1982). LSU Historical Dissertations and Theses. 3777.