Stephen McAdam

Quintasymptotic primes and four results of Schenzel

Let I be an ideal in a Noetherian ring R, let (I)a be the integral closure of I, and let S be a multiplicatively closed subset of R. Let T1, T2, and T3 be the topologies given by the filtrations {In RS ∩ R | n ≥ 1}, {In | n ≥ 1}, and {(In)a | n ≥ 1}. We g results due to Schenzel, characterizing when T1 is either equivalent or linearly equivalent to either of T2 or T3. The characterizations involve the sets of essential primes of I, quintessential primes of I, asymptotic primes of I, and quintasymptotic primes of I.

Prime divisors and divisorial ideals

Abstract Let I1,…,Ig be regular ideals in a Noetherian ring R. Then it is shown that there exist positive integers k1,…,kg such that (I1n1 + m1…Igng + mg):(I1m1…Igmg) = I1n1…Igng for all ni≥ki (i = 1,…,g) and for all nonnegative integers m1,...,mg. Using this, it is shown that if Δ is a multiplicatively closed set of nonzero ideals of R that satisfies certain hypotheses, then the sets Ass ( R (I 1 n 1 …I g n g ) ) equal for all large positive integers n1,...,ng. Also, if R is locally analytically unramified, then some related results for general sets Δ are proved.

Integrally Closed Projectively Equivalent Ideals

In this paper, I will be a regular ideal in a Noetherian ring R and Ī will denote the integral closure of I. If J is another ideal of I. and J are said to be projectively equivalent if there are positive integers n and m with \(\overline {{I^n}} \, = \,\overline {{J^m}} \). Projective equivalence was introduced by Samuel in [S], and further developed by Nagata in [N]. The concept of projective equivalence is rather interesting, but there appears to be much about it which is not known. We hope to dispel some of the darkness. We study the set of integrally closed ideals which are projectively equivalent to I, and show that this set is rather well behaved. For instance, it is linearly ordered by inclusion, and ”eventually periodic”, a phrase we will make more precise in the next paragraph. An interesting corollary to our work is that there is a fixed positive integer d such that for any ideal J projectively equivalent to I, there is a positive integer n with \(\overline {{I^n}} {\mkern 1mu} = {\mkern 1mu} \overline {{J^d}} \).

On the asymptotic cograde of an ideal

Abstract Let I be an ideal in a local ring ( R , M ) and let s ( I ) be the asymptotic cograde of I (= the maximum length of an asymptotic sequence over I ). Rees has shown that s ( I ) ⩽ altitude R − l ( I ), where l ( I ) is the analytic spread of I . We prove several more bounds on s ( I ): (i) s ( I ) ⩽ min{little depth P ; P is an asymptotic prime divisor of I ; (ii) s(I) ⩽ grade ∗ M I (= the maximum length of an asymptotic sequence contained in M I ) ; (iii) s(I) ⩽ grade ∗ M- grade ∗ I ; (iv) grade ∗ M − l(I) ⩽ s(I) ; and. (v) if R is quasi-unmixed, then grade R I n ⩽ s(I) for all large n . Also, it is shown that the asymptotic prime divisors of ( I , b 1 ,…, b s ) R contain those of I , when b 1 ,…, b s are an asymptotic sequence over I , and that the residue classes modulo I of such a sequence are an asymptotic sequence in R I . Finally, several sufficient conditions for l ( I ) to be equal to l ( I p ) for some asymptotic prime divisor P of I are given.

Essential prime divisors and projectively equivalent ideals

Let Z be an ideal in a Noetherian ring R. We concern ourselves with the essential prime divisors of Z, an interesting subset of Ass R/P, for all large n. We first take Z = bR with b a regular element of R. We show that there is a ring T, with R c TE R,, such that T is a finite R-module and the essential primes of bT are exactly the prime divisors of bT. We next consider an arbitrary ideal Z, and apply our principal arguments to the element u in the Kees ring of Z. We thereby deduce that there is an idcal J projectively equivalent to Z, such that the set of essential primes of Z equals the set U Ass R/J”, over II = 1, 2, 3 ,.... Notation. Let Z be an ideal in a Noetherian ring R. We will use W (or

UNIQUE FACTORIZATION OF MONIC POLYNOMIALS

Let R be a commutative integral domain with 1. It is trivial to see that in the polynomial ring R[X], any nonconstant monic polynomial can be factored into a product of nonconstant monic polynomials which are irreducible in R[X]. However, in an arbitrary domain, such a factorization need not be unique. We show uniqueness occurs exactly when R is integrally closed.

Filtrations, rees rings, and ideal transforms

Let I be an ideal, and let f = {Kn|n ≥ 0 } be a filtration of the Noetherian ring R, such that In ⊆ Kn for all n ≥ 0. We study when the Rees ring R(f) is either finite or integral over the Rees ring R(I), for two types of filtrations f which have recently drawn interest. If I and J are ideals in R, and if m(n) is the least power of J such that (In : Jm(n) + 1), we show that the function m(n) is eventually non-decreasing. For J regular, we characterize when it is eventually constant.

A noetherian example

Kaplansky asked if in a Noetherian domain the intersection of two height 2 primes must contain a non-zero prime. This paper presents a counterexample. Some positive results are given in [2]. The construction in the example proper considerably simpli-fies the argument of [3-Theorem 2.5]. We assume familiarity with [1, Section 1-5].

Deep decompositions of modules

A decomposition of an R-module into submodulesM = +M α, is called deep if for every submodule H of M, we have H = +(H ∩ M α). We characterize when deep decompositions exist. We then show that M ≃ +MP (over all maximal ideals P) if and only if R/(Ann m) is a finite direct sum of quasi-local rings for all 0 ≠ m ∈ M. We also show this decomposition is deep.

Bounds related to projective equivalence classes of ideals

Let Z be a regular ideal in Noetherian ring R, and let (I), denote the integral closure of I. A situation which is often of interest is when there is an integer b(Z) 2 0 such that (I” + b(‘))a E I” for all n 2 1. (For instance, in an analytically unramified local ring, such a b(Z) exists for all I.) Now recall that the ideal H is projectively equivalent to I if there are positive integers r and s with (H’), = (I”),. In Section 1, we make the easy observation that the existence of b(Z) as above is an invariant of the projective equivalence class of I. That is, if such a b(Z) exists, and if the ideal H is projectively equivalent to Z, then there is a b(H) with (H” + bCH))n _C H” for all n 2 1. Of course it may happen that b(H) # b(Z). The main purpose of Section 1 is to study the stronger condition that there exist a fixed integer b 3 0 such that for all ideals H projectively equivalent to Z, (H” + ‘), c H” for all n > 1. We make use of a key result, namely that given Z, there exists an integer d such that for any ideal H projectively equivalent to Z, one can write (Hd), = (P), for some s. Let s 2 1 be an integer, and let the ideal K be a reduction of I”. This means there is an integer e 20 (depending on s and K) with K(Z’)’ = (Z’)‘+ ‘. In Section 2, we discuss a situation in which e can in fact