William Heinzer

On the uniqueness of the coefficient ring in a polynomial ring

If k is a field and X and Y are indeterminates then the statement “consider R = Iz[X, Y] as a polynomial ring in one variable” is ambiguous, for there arc infinitely many possible choices for the ring of coefficients (e.g., If A, = k[X f Y”] then A,[Y] == B,,[Y] 7: R but A,, f J,, if m # n). On the other hand, if Z denotes the integers then the polynomial ring Z[X] has a unique subring over which it is a polynomial ring. This investigation began with our consideration of the first of these examples. In fact, Coleman had asked: If k is a field, then although k[X, 1’1 can be written as a polynomial ring in many different ways, is it true that all of the possible coefficient rings are isomorphic? That is, if T is transcendental over d and 4[T] == k[X, Y], is A a polynomial ring over k ? We found that this is indeed the case (see our (2.8)).We next proved the following: If il is a one dimensional afine domain over a

Integral domains in which each non-zero ideal is divisorial

eZd and B is a ring such that A [-Xl = B[ E;] zs an equality of polynomial rings, then either A ~ B or there is afield k such that each of A and B is a polynomial +zg in one variabZe over k. This is a corollary of (3.3) in the present paper. Our (7.7) sketches a version of the original proof. In studying this argument, we found that there were implicit in it techniques for investigating the following general question: Suppose A and B are commutative rings with identity and the polynomial rings 4 [X, , . . . , X,,] and B[E; ,..., I’,] are isomorphic, how are A and B related? Are A and B isomorphic? In particular, when does the given isomorphism take i4 onto B ? This study is mainly centered on the latter portion of the question. We are concerned almost entirely with domains It is convenient to use the following terminology which is modeled after that

The ratliff-rush ideals in a noetherian ring

Let D be an integral domain with identity having quotient field K . A non-zero fractional ideal F of D is said to be divisorial if F is an intersection of principal fractional ideals of D [4; 2]. Equivalently, F is divisorial if there is a non-zero fractional ideal E of D such that Divisorial ideals arose in the investigations of Van der Waerden, Artin, and Krull in the 1930s and were called v -ideals by Krull [9; 118]. The concept has played an important role in the development of multiplicative ideal theory.

The radical trace property

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

M-Canonical ideals in integral domains

or II-’ E Spec(R) [4, Prop. 2.41. Using this formulation of TP, Fontana, Huckaba and Papick gave characterizations for Noetherian TP domains and a special class of Priifer TP domains [4]. It is our goal to extend this work by studying RTP (radical trace property) domains, i.e., domains satisfying the condition that ZZ- ’ =

Products of commutative rings and zero-dimensionality

We prove that a Priifer domain R has an m-canonical ideal J, that is, an ideal I such that J: (I: J) = J for every ideal J of R, if and only if R is h-local with only finitely many maximal ideals that are not finitely generated; moreover, if these conditions are satisfied, then the product of the non-finitely generated maximal ideals is an m-canonical ideal of R

The Laskerian property in commutative rings

If R is a Noetherian ring and n is a positive integer, then there are only finitely many ideals I of R such that the residue class ring R/I has cardinality ≥ n. If R has Noetherian spectrum, then the preceding statement holds for prime ideals of R. Motivated by this, we consider the dimension of an infinite product of zero-dimensional commutative rings. Such a product must be either zero-dimensional or infinite-dimensional. We consider the structure of rings for which each subring is zero-dimensional and properties of rings that are directed union of Artinian subrings

Multiplicative Ideal Theory in Commutative Algebra

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.

Formal fibers and birational extensions

For over forty years, Robert Gilmer’s numerous articles and books have had a tremendous impact on research in commutative algebra. It is not an exaggeration to say that most articles published today in non-Noetherian ring theory, and some in Noetherian ring theory as well, originated in a topic that Gilmer either initiated or enriched by his work. This volume, a tribute to his work, consists of twenty-four articles authored by Robert Gilmer’s most prominent students and followers. These articles combine surveys of past work by Gilmer and others, recent results which have never before seen print, open problems, and extensive bibliographies. In a concluding article, Robert Gilmer points out directions for future research, highlighting the open problems in the areas he considers of importance. Robert Gilmer’s article is followed by the complete list of his published works, his mathematical genealogical tree, information on the writing of his four books, and reminiscences about Robert Gilmer’s contributions to the stimulating research environment in commutative algebra at Florida State in the middle 1960s. The entire collection provides an in-depth overview of the topics of research in a significant and large area of commutative algebra.

Strongly irreducible ideals of a commutative ring

Suppose (i?, m) is a local Noetherian domain with quotient field K and m-adic completion R . It is well known that the fibers of the morphism Spec(i?) —* Spec(R), i.e., the formal fibers of R, encode important information about the structure of R. Perhaps the most important condition in Grothendiecks definition of R being excellent is that the formal fibers of R be geometrically regular. Indeed, a local Noetherian ring is excellent provided it is universally catenary and has geometrically regular formal fibers [G, (7.8.3), page 214]. But the structure of the formal fibers of R is often difficult to determine. We are interested here in bringing out the interrelatedness of properties of the generic formal fiber of R with the existence of certain local Noetherian domains C birationally dominating R and having C / m C is a finite i?-module. The possible dimensions of the formal fibers of a local Noetherian ring are considered in [Ma2] and in [R3]. Following Matsumura in [Ma2], we use a(A) to denote the maximal dimension of a formal fiber of a local Noetherian ring A If (/?, m) is a local Noetherian domain with quotient field K and m-adic completion R, then Matsumura shows [Ma2, Corollary 1, page 262] that a(R) is the dimension of the generic formal fiber R[K\. He also observes that if R is of positive dimension, then a(R) r > 1 are integers, and xu... >xn are indeterminates, then two interesting examples considered in [Ma2] are the rings