Examples Built with D+M, A+XB[X] and other Pullback Constructions

Abstract

It is likely that many of the readers of this article first encountered a study of pairs of domains which have a common nonzero ideal in the exercises of R. Gilmer’s book on multiplicative ideal theory [37] (or [38]). Others may have encountered them in Appendix 2 of the original Queen’s Notes version of the same book [36], or in A. Seidenberg’s second paper on the dimension of polynomial rings [53]. Basically in all three, the concentration is on first beginning with a valuation domain V which can be written in the form K + M, then considering subrings of V which are of the form D + M where D is a domain which is contained in K. One use for such a construction is to give examples of valuation domains of larger and larger dimensions. For example, the discrete rank one valuation domain V = K[x](X) can also be written as V = K + xK[x](X) If (by chance or construction) K is equal to F(Y) for some field F and indeterminate y, then W = F + YF[Y](Y) is a discrete rank one valuation domain with quotient field K and W + x K[x](X) is a discrete rank two valuation domain with the same quotient field as V, namely, K(x). In [53], the purpose is to show that for each pair of positive integers n and m where n + l<m<2n + l, there is an integrally closed quasilocal domain R such that dim(R) = n and dim(R[x]) = m. Through the years, many authors have used this “classical” D + M construction to construct integral domains with various desired and/or undesired properties.