Projectively full ideals in Noetherian rings (II)

Abstract

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.

Reductions of ideals in local rings with finite residue fields

“…By considering unramified finite integral extensions we have been able to extend results obtained in the papers [2] and [3] in the case of infinite residue fields to the case of finite residue fields. The main result here, Theorem 3.1, gives another instance where some desirable behavior in a local ring (R, M ) which usually depends on R/M being infinite can be recovered in the finite residue field case by passing to a finite free local unramified extension ring S of R. We present in Theorem 4.1 an application of Theorem 3.1.…”

Projectively full radical ideals in integral extension rings

“…, e n of I is equal to one, then I is projectively full, by [1, (4.10)]. 3 If there exists an ideal K ∈ P(I ) whose Rees integers have greatest common divisor equal to one, then K and P(I ) are projectively full. Since the ordered sets of Rees integers of I and K are proportional, by [11, Proposition 2.10], and since P(I ) = P(K) is linearly ordered by inclusion, by [11,Corollary 2.4], necessarily K is the largest ideal in P(I ).…”

“…This definition first appeared in [1]. A number of results about, and examples of, projectively full ideals are given in [1][2][3][4]. Several characterizations of such ideals are given in [1, (4.11) and (4.12)].…”

Bases of ideals and Rees valuation rings

“…Recall that an ideal J is projectively equivalent to I if (I m ) a = ( J n ) a for some positive integers m and n. In addition to the papers [2][3][4] and the references listed there, further interesting results concerning projective equivalence can be found in [6, Proposition 2.1], [7][8][9]. We also use the following definition, see [10] and [20, p. 111].…”