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

• Thomas G. Lucas
Chapter
Part of the Mathematics and Its Applications book series (MAIA, volume 520)

## 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.

## Keywords

Maximal Ideal Integral Closure Valuation Domain Mori Domain Quotient Field
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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