David Lantz

The ratliff-rush ideals in a noetherian ring

Ratliff and Rush show in particular that Ĩ is the largest ideal for which, for sufficiently large positive integers n, (Ĩ) = I and hence that ̃̃ I = Ĩ. We call regular ideals I for which I = Ĩ Ratliff–Rush ideals, and we call Ĩ the Ratliff–Rush ideal associated with I. It is easy to see that an element a of I : I is integral over I, in the sense that there is an equation of the form a + b1a k−1 + . . . + bk = 0, where bi ∈ I for i = 1, . . . , k. Therefore, the ideal Ĩ is always between I and the integral closure I ′ of I, and hence integrally closed ideals are Ratliff–Rush ideals. Ratliff and Rush observe [RR, (2.3.4)] that the powers of an invertible ideal are Ratliff–Rush ideals, so any principal ideal generated by a nonzerodivisor is a Ratliff–Rush ideal. They also prove the interesting fact that for any regular ideal I of R, there is a positive integer n such that for all k ≥ n, Ĩk = I [RR, (2.3.2)], i.e., all sufficiently high powers of a regular ideal are Ratliff–Rush. A regular ideal I is always a reduction of its associated Ratliff–Rush ideal Ĩ, in the sense that I(Ĩ) = (Ĩ) for some positive integer n. For the basic facts on reductions and reduction numbers of ideals, we refer the reader to [NR], [H1], and [H2]. In particular, if there is an element a of an ideal I for which aI = I then aR is called a principal reduction of I and the smallest n for which this equation holds is called the reduction number of I. We will call a regular ideal I stable iff it has a principal reduction with reduction number at most one, i.e., iff there is an element a of I for which

The Laskerian property in commutative rings

1. INTRODUCTION Primary decomposition is a venerable tool in commutative algebra; indeed, Emmy Noether studied rings with the ascending chain condition on ideals because primary decomposition was available there [9 J. Though many results for which it was once used are now proved by other means, primary decomposition itself is still finding new applications [ 15, 161, and provides an often informative representation of ideals [2]. In this paper we study the class of rings (always commutative with unity) in which primary decom- position holds, and related classes. Recall: DEFINITION. Let M be a finitely generated module over ring R. (1) A submodule N is primary if, for any r in R and m in M whose product rm is in N, either m E N or some power rk of r satisfies rkM G N. It is strongly primary if, in addition, the radical P = fl= {r E R : rkM L N for some k} has a power Pk which satisfies PkM E N. (2) M is a (strongly) Laskerian module if every submodule of M is an intersection of a finite number of (strongly) primary submodules. (3) M is a ZD module if, for every submodule N of M, the set Z,(M/N) = {r E R: rm E N for some m E M\N} of zero divisors on M/N in R is the union of a finite number prime ideals in R. Of course, a ring is Laskerian, or strongly Laskerian, or ZD, if it has the property as a module over itself. In Section 2 we prove the ascent of these properties in certain ring extensions; in particular, finite integral extensions.

Commutative rings with acc on n-generated ideals

Abstract A commutative ring with unity “has n-acc” iff every ascending chain of n-generated ideals stabilizes. This paper shows that any polynomial ring or formal power series ring over a Noetherian ring has n-acc for all n. The method involves a sufficient condition for n-acc in the quasilocal case and another for globalizing the n-acc property. Examples are given to show that n-acc does not imply (n + 1)-acc for every positive integer n and that n-acc does not behave well in general under localization, globalization, and passage to a polynomial ring. It is also noted that a Prufer domain with 2-acc is Dedekind.

Artinian modules and modules of which all proper submodules are finitely generated

William D. Weakley, in [Wl], studied modules (unitary, over a commutative ring with unity) which are not finitely generated but all of whose proper submodules are finitely generated. Weakley called such modules “almost finitely generated” (a.f.g.). He noted that an a.f.g. module has prime annihilator and that, as a faithful module over a domain, it is either torsion or torsion-free. If torsion-free, it is isomorphic as a module to the domain’s quotient field. If torsion, it is Artinian, so Weakley was led to the tools of study of Artinian modules developed by Matlis [Ml, M23 and Vamos [V]. These tools include, for a quasilocal ring (R, M), the injective envelope E,(R/M) of the residue field R/M. The purpose of the present paper is to combine results of [Wl] and [GH2] to characterize the form of a.f.g. modules and the rings which admit such modules, and to describe the a.f.g. submodules of E,(R/M). In Section 2, we refine a result of [ Wl ] to conclude that, if R is a domain and (F’, P) is a discrete rank one valuation domain between R and its quotient field K, then K/V is an a.f.g. R-module if and only if V/P has finite length as an R-module. We also note that another result of [Wl] provides an affirmative answer to a question in [GH2]: Every a.f.g.module over a domain D has the form L/N, where L, N are D-submodules of the quotient field of D; indeed, L can be taken to be the localization of D at the (prime) annihilator of the module. 201 0021-8693/U

ACCP IN POLYNOMIAL RINGS: A COUNTEREXAMPLE

3.00

The Pole Assignability Property in Polynomial Rings over GCD-Domains

We describe an example showing that ACCP need not extend from a ring to a polynomial ring over it.

Projective Lines Over One-Dimensional Semilocal Domains and Spectra of Birational Extensions

Several recent papers, among them [BSSV, Tl, T2], have considered the problem of identifying rings with the property of “pole assignability.” (The definition appears below.) In particular, Bumby, Sontag, Sussmann, and Vasconcelos [BSSV] showed that, while a polynomial ring in one indeterminate over a field has this property, the polynomial ring in two indeterminates over the reals, R[x, y], and the polynomial ring in one indeterminate over the integers, Z[x], do not have this property. Then Tannenbaum [Tl, T23 showed that the polynomial ring in two indeterminates over any field does not have this property. The purpose of this note is to unify the proofs of these two facts in results that we hope will be helpful in identifying the pole assignability property (or its absence) in other rings. Let

Exceptional Prime Divisors of Two-dimensional Local Domains

In [7], Nashier asked if the condition on a one-dimensional local domain R that each maximal ideal of the Laurent polynomial ring R[y, y -1] contracts to a maximal ideal in R[y] or in R[y -1] implies that R is Henselian. Motivated by this question, we consider the structure of the projective line Proj(R[s, t]) over a one-dimensional semilocal domain R (the projective line regarded as a topological space, or equivalently as a partially ordered set). In particular, we give an affirmative answer to Nashier’s question. (Nashier has also independently answered his question [9].) Nashier has also studied implications on the prime spectrum of the Henselian property in [8] as well as in the papers cited above.

Factorization of monic polynomials

Throughout this paper, (R, M, k) will denote a 2-dimensional excellent normal (i. e., integrally closed) local (Noetherian) domain and K its field of fractions. We consider the normal local domains between R and K that are spots over R. Any such spot lies on the blow-up of some ideal I of R and on the normalized blow-up Proj(R[It]′) (where t is an indeterminate and ′ denotes integral closure) of an ideal generated by two elements. The purpose of the present article is to study containment relations among these spots and conditions under which they can appear on the normalized blow-up of a specific ideal I; in particular, we study conditions under which I can be chosen to be M-primary.

The residue fields of a zero-dimensional ring

We prove a uniqueness result about the factorization of a monic polynomial over a general commutative ring into comaximal factors. We apply this result to address several questions raised by Steve McAdam. These questions, inspired by Hensels Lemma, concern properties of prime ideals and the factoring of monic polynomials modulo prime ideals.