Valuation Overrings Research Articles

A Classification of the Integrally Closed Rings of Polynomials containing $Z[X

We study the space of valuation overrings of Z[X] by ordering them using a constructive process. This will help in classifying the integrally closed domains between Z[X] and Q[X]. Among these, we describe the Prufer ones, the Noetherian’s, the PvMD’s, etc.

GOING-DOWN AND SEMISTAR OPERATIONS

If D ⊆ T is an extension of (commutative integral) domains and ⋆ (resp., ⋆′) is a semistar operation on D (resp., T), we define what it means for D ⊆ T to satisfy the (⋆,⋆′)-GD property. Sufficient conditions are given for (⋆,⋆′)-GD, generalizing classical sufficient conditions for GD such as flatness, openness of the contraction map of spectra and the hypotheses of the classical going-down theorem. If ⋆ is a semistar operation on a domain D, we define what it means for D to be a ⋆-GD domain, generalizing the notion of a going-down domain. In determining whether a domain D is a [Formula: see text] domain, the domain extensions T of D for which [Formula: see text] is tested can be the [Formula: see text]-valuation overrings of D, the simple overrings of D, or all T. P ⋆ MD s are characterized as the [Formula: see text]-treed (resp., [Formula: see text]) domains D which are [Formula: see text]-finite conductor domains such that [Formula: see text] is integrally closed. Several characterizations are given of the [Formula: see text]-Noetherian domains D of [Formula: see text]-dimension 1 in terms of the behavior of the (⋆,⋆′)-linked overrings of D and the ⋆-Nagata rings Na(D,⋆).

Pseudo-almost valuation domains are quasilocal going-down domains, but not conversely

Ayman Badawi has recently introduced the PAVDs, a class of (commutative integral) domains which is found strictly between the class of APVDs (“almost pseudo valuation domains”) and that of the (necessarily quasilocal) domains having a linearly ordered prime spectrum. It is known that the latter class strictly contains the class of quasilocal going-down domains; it is proved that the class of quasilocal going-down domains strictly contains the class of PAVDs. Consequently, each seminormal PAVD is a divided domain. Moreover, for each n, 1 ≤ n ≤ ∞, an example is constructed of a divided domain (necessarily a quasilocal going-down domain) of Krull dimension n which is not a PAVD. Keywords Pseudo-almost valuation domain, Prime ideal, Going-down domain, Divided domain, Quasilocal, Valuation overring, Root extension, Seminormal, D+M construction, Krull dimension Mathematics Subject Classification (2000) Primary 13B24, 13G05, Secondary 13A15, 13F05

Overrings of two-dimensional Noetherian domains representable by Noetherian spaces of valuation rings

Let D be a Noetherian domain of Krull dimension 2, and let H ⊆ R be integrally closed overrings of D . We examine when H can be represented in the form H = ( ⋂ V ∈ Σ V ) ∩ R , with Σ a Noetherian subspace of the Zariski–Riemann space of the quotient field of D . We characterize also the special case in which Σ can be chosen to be a finite character collection of valuation overrings of D .

Irredundant intersections of valuation overrings of two-dimensional Noetherian domains

Let D be a two-dimensional Noetherian domain, let R be an overring of D, and let Σ and Γ be collections of valuation overrings of D. We consider circumstances under which ( ⋂ V ∈ Σ V ) ∩ R = ( ⋂ W ∈ Γ W ) ∩ R implies that Σ = Γ . We show that if R is integrally closed, these representations are “strongly” irredundant, and every member of Σ ∪ Γ has Krull dimension 2, then Σ = Γ . If in addition Σ and Γ are Noetherian subspaces of the Zariski–Riemann space of the quotient field of D (e.g. if Σ and Γ have finite character), then the restriction that the members of Σ ∪ Γ have Krull dimension 2 can be omitted. An example shows that these results do not extend to overrings of three-dimensional Noetherian domains.

Universal mapping properties of some pseudovaluation domains and related quasilocal domains

If (R, M) and (S, N) are quasilocal (commutative integral) domains and f : R → S is a (unital) ring homomorphism, then f is said to be a strong local homomorphism (resp., radical local homomorphism) if f(M) = N (resp., f(M)⊆N and for each x ∈ N, there exists a positive integer t such that xt ∈ f(M)). It is known that if f : R → S is a strong local homomorphism where R is a pseudovaluation domain that is not a field and S is a valuation domain that is not a field, then f factors via a unique strong local homomorphism through the inclusion map iR from R to its canonically associated valuation overring (M : M). Analogues of this result are obtained which delete the conditions that R and S are not fields, thus obtaining new characterizations of when iR is integral or radicial. Further analogues are obtained in which the “pseudovaluation domain that is not a field” condition is replaced by the APVDs of Badawi‐Houston and the “strong local homomorphism” conditions are replaced by “radical local homomorphism.”

POWERFUL IDEALS, STRONGLY PRIMARY IDEALS, ALMOST PSEUDO-VALUATION DOMAINS, AND CONDUCIVE DOMAINS

ABSTRACT Let R be a domain with quotient field K, and let I be an ideal of R. We say that I is powerful (strongly primary) if whenever and , we have or ( or for some ). We show that an ideal with either of these properties is comparable to every prime ideal of R, that an ideal is strongly primary it is a primary ideal in some valuation overring of R, and that R admits a powerful ideal R admits a strongly primary ideal R is conducive in the sense of Dobbs-Fedder. Finally, we study domains each of whose prime ideals is strongly primary.

Integral domains with almost integral proper overrings

Let R be an integral domain with quotient field K. By a proper nontrivial overring of R is meant a ring contained properly between R and K. It is proved that each proper nontrivial overring of R is almost integral over R if and only if either R is pointwise non-Archimedean (also known as anti-Archimedean) or R is a G-domain such that the complete integral closure of R is the unique one-dimensional valuation overring of R. This condition is also characterized in the class of seminormal domains by “R is either pointwise non-Archimedean or a maximized domain”; in the class of conducive domains (where the “maximized” part of the assertion can be replaced by “R is a G-domain”); in the class of Prufer domains (where the “maximized” part of the assertion can be replaced by “Spec (R) is pinched at a height 1 prime”); and in the classes of Archimedean and of Noetherian domains. In particular, each proper nontrivial overring of a valuation domain R is almost integral over R. On the other hand, if R is either Noetherian or one-dimensional, then no proper nontrivial overring of R can be both a ring of fractions of R and an almost integral extension of R.

Mori domains of integer-valued polynomials

Let D be a domain with quotient field K. We investigate conditions under which the ring Int(D)={f∈K[X] | f(D)⊆D} of integer-valued polynomials over D is a Mori domain. In particular, we show that if D is a pseudo-valuation domain with finite residue field such that the associated valuation overring is rank one discrete and has infinite residue field, then Int( D) is a Mori domain with Int( D)≠ D[ X]. Finally, we investigate the class group of a Mori domain of integer-valued polynomials, showing, in the case just mentioned, that Cl(Int( D)) is generated by the classes of the t-maximal uppers to zero.

On Mockor's Question

For certain classes of Prüfer domains A, we study the completion Á,T of A with respect to the supremum topology T=sup{Tw|w∈Ω}, where Ω is the family of nontrivial valuations on the quotient field which are nonnegative on A and Tw is a topology induced by a valuation w∈Ω. It is shown that the concepts “SFT Prüfer domain” and “generalized Dedekind domain” are the same. We show that if E is the ring of entire functions, then Ê,T is a Bezout ring which is not a T̂-Prüfer ring, and if A is an SFT Prüfer domain, then Á,T is a Prüfer ring under a certain condition. We also show that under the same conditions as above, Á,T is a T̂-Prüfer ring if and only if the number of independent valuation overrings of A is finite. In particular, if A is a Dedekind domain (resp., h-local Prüfer domain), then Á,T is a T̂-Prüfer ring if and only if A has only finitely many prime ideals (resp., maximal ideals). These provide an answer to Mockor's question.

A going-up theorem for arbitrary chains of prime ideals

We prove that if an extension R ⊆ T of commutative rings satisfies the going-up property (for instance, if T is an integral extension of R), then any increasing chain of prime ideals of R (indexed by an arbitrary linearly ordered set) is covered by some corresponding chain of prime ideals of T. As a corollary, we recover the recent result of Kang and Oh that any such chain of prime ideals of an integral domain D is covered by a corresponding chain in some valuation overring of D.

Discrete valuation overrings of Noetherian domains

We show that, given a chain $0=P_0\subset P_1\subset \dotsb \subset P_n$ of prime ideals in a Noetherian domain $R$, there exist a finitely generated overring $T$ of $R$ and a saturated chain of primes in $T$ contracting term by term to the given chain. We further show that there is a discrete rank $n$ valuation overring of $R$ whose primes contract to those of the given chain.

The “Defektsatz” for central simple algebras

LetQQbe a central simple algebra finite-dimensional over its centerFFand letVVbe a valuation ring ofFF. ThenVVhas an extension toQQ, i.e., there exists a Dubrovin valuation ringBBofQQsatisfyingV=F∩BV= F \cap B. Generally, the number of extensions ofVVtoQQis not finite and therefore the so-called intersection property of Dubrovin valuation ringsB1,…,Bn{B_1}, \ldots ,{B_n}is introduced. This property is defined in terms of the prime ideals and the valuation overrings of the intersectionB1∩⋯∩Bn{B_1} \cap \cdots \; \cap {B_n}. It is shown that there exists a uniquely determined natural numbernndepending only onVVand having the following property: IfB1,…,Bk{B_1}, \ldots ,{B_k}are extensions ofVVhaving the intersection property thenk≤nk \leq nandk=nk= nholds if and only ifB1∩⋯∩Bk{B_1} \cap \cdots \cap {B_k}is integral overVV. Letnnbe the extension number ofVVtoQQ. There exist extensionsB1,⋯,Bn{B_1}, \cdots ,{B_n}ofVVhaving the intersection property and ifR1,…,Rn{R_1}, \ldots ,{R_n}are also extensions ofVVhaving the intersection property thenB1∩⋯∩Bn{B_1} \cap \cdots \cap {B_n}andR1∩⋯∩Rn{R_1} \cap \cdots \cap {R_n}are conjugate. The main result regarding the extension number is the Defektsatz:[Q:F]=fB(Q/F)eB(Q/F)n2pd[Q:F]= {f_B}(Q/F){e_B}(Q/F){n^2}{p^d}, wherefB(Q/F){f_B}(Q/F)is the residue degree,eB(Q/F){e_B}(Q/F)the ramification index,nnthe extension number,p=char⁡(V/J(V))p = \operatorname {char}(V/J(V)), anddda natural number.

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 Prüfer domain, in which case P = PR p . Consequences include pullback characterizations of the seminormal (resp., Prüfer) 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.

Analogues of a theorem of Cohen for overrings

An integral domain R has the finite [finite presentation] overring property, if every overring of R in the quotient field K of R is a finitely generated [finitely presented] R-algebra. Papick in [24] (Proposition 23) has given a characterization of one-dimensional domains with the finite presentation overring property. In this paper, we remove the restriction on the dimension. Thri is achieved in Theorem 18 using a characterization of domains with the finite overring property (Theorem 14). An important contribution in obtaining these results comes from a consideration of an analogue for overrings of the following theorem of Cohen [17, Theorem 71. Let 1 be an ideal of a commutative unitary ring R. Assume that I is not finitely generated and is maximal with respect to this property. Then I is a prime ideal of R. Studies on overrings of integral domains [ 12,13,14,18,28] suggest that, in multiplicative domains, overrings of domains play a role analogous to ideals in commutative rings and that valuation overrings may be thought of as analogues of prime ideals. Looking for an analogue of the above theorem of Cohen, we are led to introduce the following definition: Let R be a G-domain, with quotient field K. We say that an overring S of R in K is a Cohen overring of R, if S is not a finitely generated R-algebra and S is maximal with respect to this property. We show that Cohen overrings are always local rings (Proposition 2). We exhibit examples showing that they may not be valuation rings. However, we point out that there are

Pairs of Rings with the Same Prime Ideals

There are numerous instances in which the partners in an extension of commutative rings R ⊂ T have the same prime ideals, i.e., in which Spec(R) = Spec(T). Although this equality is intended to be taken set-theoretically, the identification easily extends to the corresponding spaces endowed with their Zariski topologies (see Proposition 3.5(a)), but of course need not extend to an identification of Spec(R) and Spec(T) as affine schemes. Perhaps the most striking recent illustration of this phenomenon arises from the work of Hedstrom and Houston [14] in which R is a pseudo-valuation domain and T is a suitable valuation overring. Other examples may be found by means of the D + M construction, either in its traditional form [12, p. 560] or in the generalized situation introduced by Brewer and Rutter [5].

Notes on a new chain condition for prime ideals

A chain condition intermediate to the catenary property and the chain condition for prime ideals (c.c.) is studied. Like the c.c., the condition is inherited from a semi-local domain R by integral extension domains, by local quotient domains, and by factor domains, and a semi-local ring that satisfies the condition is catenary. (Unlike the c.c., none of these statements is true when R is not semi-local.) A number of characterizations of a semi-local domain that satisfies the condition are given in terms of: integral (respectively, algebraic, transcendental) extension domains, Henselizations, completions, Rees rings, associated graded rings and certain discrete valuation over-rings. Then four of the catenary chain conjectures are characterized in terms of this condition.

On Arnold’s formula for the dimension of a polynomial ring

If R R is a commutative integral domain with quotient field K K and x 1 , … , x n {x_1}, \ldots ,{x_n} are indeterminates, then there exist θ 1 , … , θ n {\theta _1}, \ldots ,{\theta _n} in K K such that dim ⁡ R [ x 1 , … , x n ] = n + dim ⁡ R [ θ 1 , … , θ n ] \dim R[{x_1}, \ldots ,{x_n}] = n + \dim R[{\theta _1}, \ldots ,{\theta _n}] .

On Going-Down for Simple Overrings. III

Theorem 1. Let $R$ be an integral domain with quotient field $K$. The following three conditions are equivalent: (a) $R \subset R[u]$ satisfies going-down $(GD)$ for each $u$ in $K$; (b) $R \subset V$ satisfies $GD$ for each valuation overring $V$ of $R$; (c) $R \subset S$ satisfies $GD$ for each domain $S$ containing $R$. If (d) is the condition obtained by restricting the domains $S$ in $({\text {c)}}$ to be overrings of $R$, then $({\text {a)}} \Leftrightarrow {\text {(d)}}$ has been proved in case $R$ is Krull or integrally closed finite-conductor (e.g., pseudo-Bézout) or Noetherian. Theorem 2. Let $R \subset T$ be domains such that either $\operatorname {Spec} (R)$ or $\operatorname {Spec} (T)$, as a poset under inclusion, is a tree. If $R \subset R[u,v]$ satisfies $GD$ for each $u$ and $v$ in $T$, then $R \subset T$ satisfies $GD$.

When $(D[[X]])\sb{P[[X]]}$ is a valuation ring

Let D be an integral domain with identity and let K denote the quotient field of D. If P is a prime ideal of D denote by P[[X]] that prime ideal of D[[X]] consisting of all those formal power series each of whose coefficients belongs to P. In this paper the following question is considered: When is (D[[X]h)r[[x]] a valuation ring? Our main theorem states that if (D[[X]])p[[x]] is a valuation ring, then Dp must be a rank one discrete valuation ring. Moreover, we show that if Dp is a rank one discrete valuation ring and if PD[[X]]=P[[X]], then (D[[X]])p[[x]] is a valuation ring. We also give an example to show that (D[[X]])Prrx]] need not be a valuation ring when Dp is rank one discrete. 1. Main theorem. Let V be a rank one valuation ring with quotient field K and let M denote the maximal ideal of V. If v is a valuation on K associated with V, then we may take the value group of v to be a subgroup of the additive group of real numbers. Forf= >j0afXi, define v*(f)= inf0o,.:,{v(ft)}. Equipped with this terminology, we have LEMMA 1. v* is a valuation on V[[X]]. Moreover, MV[[X]] is a prime ideal of V[[X]] and (V[[X]])_jsv[[x]] is the valuation ring associated with v*. PROOF. Letf= Z;tOfiX?, g = T!o giXi E V[ [X]]. It is straightforward to verify that v*(f+g)>min{v*(f), v*(g)}. To show that v*(fg)= v*(f)+v*(g), it suffices to show that v*(fg)?v*(f)+v*(g). We first consider the case in which there exist coefficients fr and g, of f and g respectively, with the property that v*(f)=v(fr) and v*(g)=v(g,). If r and s are minimal with this property, then v ( fig) = v*(f) + v*(g) i+i=r+s and since (Zi+i=r+sfig) is the coefficient of Xr+s in fg, it follows that v*(fg)?v*(f)+v*(g). In particular, if v is a discrete valuation, this shows that v* is a valuation. Assuming now that v is a nondiscrete valuation, let f, g be arbitrary elements of V[[X]] and let m E M\(O). Iff '= :E?_ i i Presented to the Society, November 20, 1970 under the title Essential valuation overrings of D[[X]]; received by the editors February 29, 1972. AMS (MOS) subject classiflcations (1969). Primary 1393; Secondary 1398.