David E. Dobbs

Pairs of rings with the same prime ideals. II

Introduction. Much of [2] was devoted to studying pairs of subrings A C B of a field with the property that A and B have the same prime ideals. In this paper, we continue that investigation, but we no longer assume that A and B are comparable. Interestingly, most of the results of [2] carry over to this more general context. Besides such extensions of [2], additional motivation for the more general context comes from the need to explicate some naturally occurring examples (see Examples 2.5, 3.6, and 4.3). Section 2 begins by showing that we may reduce to the case in which R is a quasilocal domain with nonzero maximal ideal M and quotient field K. Proposition 2.3 establishes that the set C{R) of all subrings A of K with Spec(A) = Spec(/?) forms a complete semilattice. Theorem 2.4 shows that C (R) is naturally isomorphic to the complete semilattice J (A) of all subfields of the ring A — (M : M)jM. Conversely, Theorem 2.6 shows that for any commutative ring A which contains a field, J (A) may be realized as C (R) for some quasilocal domain R. In Section 3, we investigate various common ring-theoretic properties of the rings in C(R), with special emphasis on the Noetherian property. Specifically, Theorem 3.3 gives several equivalent conditions for each A G C(R) to be Noetherian; when these conditions hold, C(R) is finite. In the final section, we study the semilattice !F (A) and give several examples that illuminate the preceding material. All rings are assumed to be commutative, with 1. Usually, R will denote a quasilocal domain with nonzero maximal ideal M and quotient field K, and k will be the prime subfield of R/M. As usual, we write

On Going Down For Simple Overrings II

This paper deals with two themes involving the going down (GD) behavior of the overrings of a domain R. First, as illustrated by several results in [3] and [4], hypotheses about GD often replace stronger assumptions about flatness or one-dimensionality in sets of conditions implying that R is Prufer. Second, as shown by the two-dimensional example in [4]. Remarks (iii)], some sort of finiteness condition is required in conjunction with GD hypotheses in order for R to be Prufer.

Universally catenarian integral domains

Abstract The rings of the title are the (not necessarily Noetherian) integral domains R such that R [ X 1 , …, X n ] is catenarian for each positive integer n . It is proved that each such R must be a stably strong S -domain, in the sense of Malik-Mott. The class of all universally catenarian integral domains is characterized as the largest class of catenarian integral domains which is stable under factor domains and localizations and whose members R satisfy the altitude formula and dim V ( R ) = dim( R ). Moreover, the following theorem is given, generalizing Ratliffs result that each one-dimensional Noetherian integral domain is universally catenarian. Let R be a locally finite-dimensional going-down domain; then R is universally catenarian if and only if the integral closure of R is a Prufer domain. Other results and applications are also given.

On going down for simple overrings

This paper deals with two themes involving the going down (GD) behavior of the overrings of a domain R. First, as illustrated by several results in [3] and [4], hypotheses about GD often replace stronger assumptions about flatness or one-dimensionality in sets of conditions implying that R is Prufer. Second, as shown by the two-dimensional example in [4]. Remarks (iii)], some sort of finiteness condition is required in conjunction with GD hypotheses in order for R to be Prufer.

Conducive integral domains

This article introduces the concept of a conducive domain, that is, an integral domain each of whose overrings, apart from the quotient field, has nonzero conductor. The principal examples of conducive domains are all D + M constructions and all (pseudo-) valuation domains. Several characterizations are obtained, notably that a domain R is conducive if and only if R has a valuation overring with nonzero conductor. It is proved that if R is a conducive domain but not a field, then Spec(R) is pinched at a prime P such that the set of primes within P is linearly ordered. The converse is shown for R a Prufer domain, in which case P = PRp. Consequences include pullback characterizations of the seminormal (resp., Prufer) conducive domains. Special attention is paid to the class of conducive G-domains, with attendant interplay between “conducive” and the property of having a maximum overring.

When is

Let V be a valuation ring of the form K + M, where K is a field and M(# 0) is the maximal ideal of V. Let D be a proper subring of K. Necessary and sufficient conditions are given that the ring D + M be coherent. The condition that a given ideal of V be D + A/-flat is also characterized.

D+M

It is proved that if u is an element of a faithful algebra over a commutative ring R , then u satisfies a polynomial over R which has unit content if and only if the extension R ⊂ R [ u ] has the imcomparability property. Applications include new proofs of results of Gilmer-Hoffmann and Papick, as well as a characterization of the P-extensions introduced by Gilmer and Hoffmann.

coherent?

ABSTRACT A (unital) extension R ⊆ T of (commutative) rings is said to have FIP (respectively be a minimal extension) if there are only finitely many (respectively no) rings S such that R ⊂ S ⊂ T. Transfer results for the FIP property for extensions of Nagata rings are obtained, including the following fact: if R ⊂ T is a (module-) finite minimal ring extension, then R(X)⊂T(X) also is a (module-) finite minimal ring extension. The assertion obtained by replacing “is a (module-) finite minimal ring extension” with “has FIP” is valid if R is an infinite field but invalid if R is a finite field. A generalization of the Primitive Element Theorem is obtained by characterizing, for any field (more generally, any artinian reduced ring) R, the ring extensions R ⊆ T which have FIP; and, if R is any field K, by describing all possible structures of the (necessarily minimal) ring extensions appearing in any maximal chain of intermediate rings between K and any such T. Transfer of the FIP and “minimal extension” properties is given for certain pullbacks, with applications to constructions such as CPI-extensions. Various sufficient conditions are given for a ring extension of the form R ⊆ R[u], with u a nilpotent element, to have or not have FIP. One such result states that if R is a residually finite integral domain that is not a field and u is a nilpotent element belonging to some ring extension of R, then R ⊆ R[u] has FIP if and only if (0 : u) ≠ 0. The rings R having only finitely many unital subrings are studied, with complete characterizations being obtained in the following cases: char(R)>0; R an integral domain of characteristic 0; and R a (module-)finite extension of ℤ which is not an integral domain. In particular, a ring of the last-mentioned type has only finitely many unital subrings if and only if (ℤ:R)≠0. Some results are also given for the residually FIP property.

On INC-extensions and polynomials with unit content

Antimatter domains are defined to be the integral domains which do not have any atoms. It is proved that each integral domain can be em-bedded as a subring of some antimatter domain which is not a field. Any fragmented domain is an antimatter domain, but the converse fails in each positive Krull dimension. A detailed study is made of the passage of the“an-timatter”property between the partners within an overring extension. Special attention is given to characterizing antimatter domains in classes of valuation domains, pseudo-valuation domains, and various types of pullbacks.

On the FIP Property for Extensions of Commutative Rings

Abstract Let R be an integral domain, X(R) the abstract Riemann surface of R, and (R′)b the Kronecker function ring of the integral closure R′ of R. It is proved that there exists a homeomorphism, natural in R, between X(R) and Spec((R′)b). Ideal-theoretic and topological results are given for the extension j: R (R ′ ) b , notably that R is a Prufer domain if and only if R = R′ and j is universally going-down. It is also proved that each spectral space X is a closed spectral image of a treed spectral space Y; if X is irreducible, Y can be taken as an abstract Riemann surface.