Integral Extension Domain Research Articles

A note on Itoh (e)-valuation rings of an ideal

Let I be a regular proper ideal in a Noetherian ring R, let e≥2 be an integer, let Te=R[u,tI,u1e]′∩R[u1e,t1e] (where t is an indeterminate and u=1t), and let re=u1eTe. Then the Itoh (e)-valuation rings of I are the rings (Te/z)(p/z), where p varies over the (height one) associated prime ideals of re and z is the (unique) minimal prime ideal in Te that is contained in p. We show, among other things:(1) re is a radical ideal if and only if e is a common multiple of the Rees integers of I.(2) For each integer k≥2, there is a one-to-one correspondence between the Itoh (k)-valuation rings (V⁎,N⁎) of I and the Rees valuation rings (W,Q) of uR[u,tI]; namely, if F(u) is the quotient field of W, then V⁎ is the integral closure of W in F(u1k).(3) For each integer k≥2, if (V⁎,N⁎) and (W,Q) are corresponding valuation rings, as in (2), then V⁎ is a finite integral extension domain of W, and W and V⁎ satisfy the Fundamental Equality with no splitting. Also, if uW=Qe, and if the greatest common divisor of e and k is d, and c is the integer such that cd=k, then QV⁎=N⁎c and [(V⁎/N⁎):(W/Q)]=d. Further, if uW=Qe and k=qe is a multiple of e, then there exists a unit θe∈V⁎ such that V⁎=W[θe,u1k] is a finite free integral extension domain of W, QV⁎=N⁎q, N⁎=u1kV⁎, and [V⁎:W]=k.(4) If the Rees integers of I are all equal to e, then V⁎=W[θe] is a simple free integral extension domain of W, QV⁎=N⁎=u1eV⁎, and [V⁎:W]=e=[(V⁎/N⁎):(W/Q)].

Projective equivalence of ideals in Noetherian integral domains

Let I be a nonzero proper ideal in a Noetherian integral domain R. In this paper we establish the existence of a finite separable integral extension domain A of R and a positive integer m such that all the Rees integers of IA are equal to m. Moreover, if R has altitude one, then all the Rees integers of J = Rad ( I A ) are equal to one and the ideals J m and IA have the same integral closure. Thus Rad ( I A ) = J is a projectively full radical ideal that is projectively equivalent to IA. In particular, if R is Dedekind, then there exists a Dedekind domain A having the following properties: (i) A is a finite separable integral extension of R; and (ii) there exists a radical ideal J of A and a positive integer m such that I A = J m . In this case the extension A also has the property that for each maximal ideal N of A with I ⊆ N , the canonical inclusion R / ( N ∩ R ) ↪ A / N is an isomorphism, and the integer m is a multiple of [ A ( 0 ) : R ( 0 ) ] .

Projectively full ideals in Noetherian rings (II)

Let R be a Noetherian commutative ring with unit 1 ≠ 0 , and let I be a regular proper ideal of R. The set P ( I ) of integrally closed ideals projectively equivalent to I is linearly ordered by inclusion and discrete. There is naturally associated to I and to P ( I ) a numerical semigroup S ( I ) ; we have S ( I ) = N if and only if every element of P ( I ) is the integral closure of a power of the largest element K of P ( I ) . If this holds, the ideal K and the set P ( I ) are said to be projectively full. A special case of the main result in this paper shows that if R contains the rational number field Q , then there exists a finite free integral extension ring A of R such that P ( I A ) is projectively full. If R is an integral domain, then the integral extension A has the property that P ( ( I A + z ∗ ) / z ∗ ) is projectively full for all minimal prime ideals z ∗ in A. Therefore in the case where R is an integral domain there exists a finite integral extension domain B = A / z ∗ of R such that P ( I B ) is projectively full.

Tight closure commutes with localization in binomial rings

It is proved that tight closure commutes with localization in any domain which has a module finite extension in which tight closure is known to commute with localization. It follows that tight closure commutes with localization in rings, in particular in semigroup or toric rings. Tight closure, introduced in [HHI], is a closure operation performed on ideals in a commutative, Noetherian ring containing a field. Although it has many important applications to commutative algebra and related areas, some basic questions about tight closure remain open. One of the biggest such open problems is whether or not tight closure commutes with localization. For an exposition of tight closure, including the localization problem, see [Hu]. The purpose of this paper is give a simple proof that tight closure commutes with localization for a class of rings that includes semi-group rings and, slightly more generally, the so-called binomial rings, that is, those that are a quotient of a finitely generated algebra by an ideal generated by binomials. Although this is a modest class of rings, the result here includes coordinate rings of all affine toric varieties, and is general enough to substantially improve previous results of a number a mathematicians, including Moty Katzman, Will Traves, Keith Pardue and Karen Chandler, Irena Swanson and myself. Throughout this note, R denotes a Noetherian commutative ring of prime characteristic. We say that tight closure commutes with localization in R if, for all ideals I in R and all multiplicative systems U in R, I*R[U-1] = (IR[U-1])*. The main result is the following: Theorem. Let R be a ring with the following property: for each minimal prime P of R, the quotient R/P has an integral extension domain in which tight closure commutes with localization. Then tight closure commutes with localization in R. Lemma 1. If tight closure commutes with localization in R/P for each minimal prime P of R, then tight closure commutes with localization in R. Received by the editors January 11, 1999 and, in revised form, May 15, 1999. 1991 Mathematics Subject Classification. Primary 13A35.

Strongly Comaximizable Primes

Let P be a prime ideal in an integral domain R with R lying between some Noetherian domain H and the integral closure of H. We call P strongly comaximizable if for every integer m ≥ 1, there exists a finitely generated integral extension domain T of R such that T has exactly m primes lying over P, and those m primes are pairwise comaximal. We show that if P is not contained in the Jacobson radical of R, then P is strongly comaximizable. We also show that if P is not strongly comaximizable and R is either integrally closed or quasi-local, then P satisfies a certain "Henselian like" property.

Comaximizable Primes

Let ${P_1}, \ldots ,{P_n}\left ( {n \geq 2} \right )$ be not necessarily distinct nonzero prime ideals in the Noetherian, but not Henselian, domain $R$. We show that there is a finitely generated integral extension domain $T$ of $R$, containing distinct, pairwise comaximal prime ideals ${Q_1}, \ldots ,{Q_n}$ lying over ${P_1}, \ldots ,{P_n}$ respectively.

Six notes on chains of prime ideals

The following topics are considered: saturated chains of prime ideals in quadratic (resp., simple, local, and level) integral extension domains of a local domain R; the heights of a prime ideal and its contraction in integral extension domains of R; and, the nonexistence of intermediate rings between R and finite integral extension domains of R that are minimal with spect to having certain properties.

Notes on three integral dependence theorems

If P is a prime ideal in an integral extension domain A of a quasi-local domain R and if F is the quotient field of R, then the following results hold when R is integrally closed: height P = height P ∩ R; altitude A p = altitude A p ∩ F; ( A p )′ ∩ F = ( A p ∩ F)′, where the prime denotes integral closure. This paper is concerned with generalizing these results to the case where R is not integrally closed.

Note on Simple Integral Extension Domains and Maximal Chains of Prime Ideals

It is shown that if R is a semi-local (Noetherian) domain, then there exists a simple integral extension domain $R[e]$ of R such that there exists a maximal chain of prime ideals of length n in some integral extension domain of R if and only if there exists a maximal chain of prime ideals of length n in $R[e]$. An interesting existence theorem on a certain type of height one prime ideals in $R[X]$ follows.

Maximal chains of prime ideals in integral extension domains III

Maximal chains of prime ideals in integral extension domains III

Note on simple integral extension domains and maximal chains of prime ideals

It is shown that if R is a semi-local (Noetherian) domain, then there exists a simple integral extension domain R [ e ] R[e] of R such that there exists a maximal chain of prime ideals of length n in some integral extension domain of R if and only if there exists a maximal chain of prime ideals of length n in R [ e ] R[e] . An interesting existence theorem on a certain type of height one prime ideals in R [ X ] R[X] follows.

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.

Notes on Local Integral Extension Domains

All rings in this paper are assumed to be commutative with identity, and the undefined terminology is the same as that in [3]. In 1956, in an important paper [2], M. Nagata constructed an example which showed (among other things): (i) a maximal chain of prime ideals in an integral extension domain R' of a local domain (R, M) need not contract in R to a maximal chain of prime ideals; and, (ii) a prime ideal P in R' may be such that height P < height P ∩ R. In his example, Rf was the integral closure of R and had two maximal ideals. In this paper, by using Nagata's example, we show that there exists a finite local integral extension domain of D = R[X](M,X) for which (i) and (ii) hold (see (2.8.1) and (2.10)).

Maximal Chains of Prime Ideals in Integral Extension Domains. II

Maximal Chains of Prime Ideals in Integral Extension Domains. II

Maximal Chains of Prime Ideals in Integral Extension Domains. I

Let (R, M) be a local domain, let k be a positive integer, and let Q be a prime ideal in ${R_k} = R[{X_1}, \ldots ,{X_k}]$ such that $M{R_k} \subset Q$. Then the following statements are equivalent: (1) There exists an integral extension domain of R which has a maximal chain of prime ideals of length n. (2) There exists a minimal prime ideal z in the completion of R such that depth $z = n$. (3) There exists a minimal prime ideal w in the completion of ${({R_k})_Q}$ such that depth $w = n + k - {\text {depth}}\;Q$. (4) There exists an integral extension domain of ${({R_k})_Q}$ which has a maximal chain of prime ideals of length $n + k - {\text {depth}}\;Q$. (5) There exists a maximal chain of prime ideals of length $n + k - {\text {depth}}\;Q$ in ${({R_k})_Q}$. (6) There exists a maximal chain of prime ideals of length $n + 1$ in $R{[{X_1}]_{(M,{X_1})}}$.

Maximal chains of prime ideals in integral extension domains. I

Let (R, M) be a local domain, let k be a positive integer, and let Q be a prime ideal in R k = R [ X 1 , … , X k ] {R_k} = R[{X_1}, \ldots ,{X_k}] such that M R k ⊂ Q M{R_k} \subset Q . Then the following statements are equivalent: (1) There exists an integral extension domain of R which has a maximal chain of prime ideals of length n. (2) There exists a minimal prime ideal z in the completion of R such that depth z = n z = n . (3) There exists a minimal prime ideal w in the completion of ( R k ) Q {({R_k})_Q} such that depth w = n + k − depth Q w = n + k - {\text {depth}}\;Q . (4) There exists an integral extension domain of ( R k ) Q {({R_k})_Q} which has a maximal chain of prime ideals of length n + k − depth Q n + k - {\text {depth}}\;Q . (5) There exists a maximal chain of prime ideals of length n + k − depth Q n + k - {\text {depth}}\;Q in ( R k ) Q {({R_k})_Q} . (6) There exists a maximal chain of prime ideals of length n + 1 n + 1 in R [ X 1 ] ( M , X 1 ) R{[{X_1}]_{(M,{X_1})}} .

Maximal chains of prime ideals in integral extension domains. II

Four related subjects are investigated: (1) If (L, N) is a locality over a local domain (R, M) such that N ∩ R = M N \cap R = M , and if there exists an integral extension domain of L which has a maximal chain of prime ideals of length n (for short, a mcpil n), then there exists an integral extension domain of R which has a mcpil n − trd L / R + trd ( L / N ) / ( R / M ) n - {\text {trd}}\;L/R + {\text {trd}}(L/N)/(R/M) . A refinement of the altitude inequality follows from this. (2) A condition for the converse of (1) to hold is given. (3) The class of local domains R such that there exists an integral extension domain of R which has a mcpil n if and only if there exists a mcpil n in R is studied. (4) Two new equivalences for the existence of mcpil n in an integral extension domain of a local domain are given.