| Although there exists a complete description of the prime ideals of finite direct product rings, very little is known concerning the spectrum of prime ideals of an infinite product ring. Thus our original goal was to determine the basic structure and cardinality of Spec R, where R is the direct product of the p-adic integer rings. A hint of the structure and cardinality of Spec R was provided in a paper written by M. Henriksen describing prime ideals of the ring of entire functions. Our direct product ring and the ring of entire functions have some similar properties, one specifically being they are Bezout rings. We use this fact to study chains of prime ideals in R, each chain being uniquely determined by an ultrafilter on IN. Using the I. Kaplansky criteria of immediate neighbor pairs, we determined the basic structure of Spec R to be similar to the Cantor set. To determine the cardinality of Spec R, we created constructive processes; starting with the elements of R, we construct all of Spec R. Primarily, this is accomplished through infinite trees of prime ideals, and more specifically, the union of primes from a 'branch' in a tree. The constructive processes yield different types of prime ideals, each type having certain characteristics. These types form a partition of Spec R. Additionally, we created a function from Spec R into and, assuming the continuum hypothesis, show the cardinality of SpecR is . Thus not only do we show structure and cardinality of Spec R, we have a complete description of all of Spec R.;Beyond the immediate questions answered by this paper concerning the one specific ring cited, the more important aspect is the constructive nature of many of the major proofs. Thus they give insight into the relationship between prime ideals of a direct product ring. These constructions then might be used to fully describe other infinite direct product rings. |
