92 research outputs found
Well-centered overrings of an integral domain
Let A be an integral domain with field of fractions K. We investigate the
structure of the overrings B of A (contained in K) that are well-centered on A
in the sense that each principal ideal of B is generated by an element of A. We
consider the relation of well-centeredness to the properties of flatness,
localization and sublocalization for B over A. If B = A[b] is a simple
extension of A, we prove that B is a localization of A if and only if B is flat
and well-centered over A. If the integral closure of A is a Krull domain, in
particular, if A is Noetherian, we prove that every finitely generated flat
well-centered overring of A is a localization of A. We present examples of
(non-finitely generated) flat well-centered overrings of a Dedekind domain that
are not localizations.Comment: Example 3.11 was replace
Excellent Normal Local Domains and Extensions of Krull Domains
We consider properties of extensions of Krull domains such as flatness that
involve behavior of extensions and contractions of prime ideals. Let (R,m) be
an excellent normal local domain with field of fractions K, let y be a nonzero
element in m, and let R* denote the (y)-adic completion of R. For a finite set
w of elements of yR* that are algebraically independent over R, we construct
two Krull domains: an intersection domain A that is the intersection of R* with
the field of fractions of K[w], and an approximation domain B to A. If R is
countable with dim R at least 2, we prove that there exist sets w as above such
that the extension R[w] to R*[1/y] is flat. In this case B = A is Noetherian,
but may fail to be excellent as we demonstrate with examples. We present
several theorems involving the construction. These theorems yield examples
where B is properly contained in A and A is Noetherian while B is not
Noetherian, and other examples where B = A is not Noetherian.Comment: 24 pages to appear in JPA
Generic fiber rings of mixed power series/polynomial rings
Let K be a field, m and n positive integers, and X = {x_1,...,x_n}, and Y =
{y_1,..., y_m} sets of independent variables over K. Let A be the polynomial
ring K[X] localized at (X). We prove that every prime ideal P in A^ = K[[X]]
that is maximal with respect to P\cap A = (0) has height n-1. We consider the
mixed power series/polynomial rings B := K[[X]][Y]_{(X,Y)} and C :=
K[Y]_{(Y)}[[X]]. For each prime ideal P of B^ = C that is maximal with respect
to either P \cap B = (0) or P \cap C = (0), we prove that P has height n+m-2.
We also prove that each prime ideal P of K[[X, Y]] that is maximal with respect
to P \cap K[[X]] = (0) is of height either m or n+m-2.Comment: 28 page
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