Suppose R is a commutative ring with identity. This paper deals with the problem of determining conditions under which R can be embedded in a zero-dimensional ring. Work on this question was begun by Arapovic in [1]-[3] and continued by Gilmer and Heinzer in [7],[8]. Essentially one set of conditions equivalent to embeddability is known. This set consists of two conditions (which we label as (A1) and (A2)) and is due to Arapovic; they are stated in Theorem 3.1. In Theorem 4.1 we establish a new criterion for embeddability of R. It states that R is embeddable in a zero-dimensional ring only if, for each x in R, there is a positive integer n (which may depend upon x) such that x(n) and x(n+1) have the same annihilator in R. We consider applications of this criterion both for known results concerning embeddability and for some open questions. The question of whether the named criterion is sufficient for embeddability in a zero-dimensional ring has not been resolved.

. For a subset E of an integral domain D and an integer-valued polynomial f over D , we investigate conditions under which the subsets E and f ( E ) of D determine the same integer-valued polynomials on D (this is the definition of polynomial equivalence of E and f ( E )). Our primary interest in this problem lies in the case where D is the ring of rational integers. Using work of McQuillan, the case where E is finite is resolved completely in Section 3. For E infinite we show in several cases that polynomial equivalence of E and f ( E ) implies that f is linear, but whether this is true in general for, say, D = Z is an open question.

Our work in this paper was motivated by a question raised by D. D. Anderson (see [A1], [A2]). Specifically, Anderson asked whether the condition that each of its finitely generated ideals admits a finite primary decomposition is inherited by the polynomial ring R[x] from the coefficient ring R. In exploring the question we encountered general questions concerning finite primary decomposition that seemed at least as interesting as the original question, and these answers would be helpful in attacking the original. Perhaps the most fundamental problem in this regard is that no easily applicable criteria are known for determining whether a fixed ideal of a commutative ring admits a finite primary decomposition. (Krull raised the problem of determining such criteria in [Kr, p. 12], and the problem is addressed in [GH2, Section 1] and in [F].)

We determine equivalent conditions on a commutative Artinian ring S S in order that the ideal of S [ t ] S[t] consisting of polynomials that vanish on S S should be principal. Our results correct an error in a paper of Niven and Warren.

Suppose M M is a maximal ideal of a commutative integral domain R R and that some power M n M^n of M M is finitely generated. We show that M M is finitely generated in each of the following cases: (i) M M is of height one, (ii) R R is integrally closed and ht ⁡ M = 2 \operatorname {ht} M=2 , (iii) R = K [ X ; S ~ ] R = K[X;\tilde S] is a monoid domain over a field K K , where S ~ = S ∪ { 0 } \tilde S = S \cup \{0\} is a cancellative torsion-free monoid such that ⋂ m = 1 ∞ m S = ∅ \bigcap _{m=1}^\infty mS=\emptyset , and M M is the maximal ideal ( X s : s ∈ S ) (X^s:s\in S) . We extend the above results to ideals I I of a reduced ring R R such that R / I R/I is Noetherian. We prove that a reduced ring R R is Noetherian if each prime ideal of R R has a power that is finitely generated. For each d d with 3 ≤ d ≤ ∞ 3 \le d \le \infty , we establish existence of a d d -dimensional integral domain having a nonfinitely generated maximal ideal M M of height d d such that M 2 M^2 is 3 3 -generated.

Let R be a commutative ring with identity. We consider conditions in order that there exists an embedding of R in a local ring.. This leads naturally to an examination of conditions in order that a quasilocal ring (R, m) be dominated by a local ring. This, in turn, leads to a study of extensions of the residue field of a quasilocal ring.

We investigate the structure of prime ideals of finite height in polynomial extension rings of a commutative unitary ring R. We consider the question of finite generation of such prime ideals. The valuative dimension of prime ideals of R plays an important role in our considerations. If X is an infinite set of indeterminates over R, we prove that every prime ideal of R[X] of finite height is finitely generated if and only if each P is an element of Spec(R) of finite valuative dimension is finitely generated and for each such P every finitely generated extension domain of R/P is finitely presented. We prove that an integrally closed domain D with the property that every prime ideal of finite height of D[X] is finitely generated is a Prufer v-multiplication domain, and that if D also satisfies d.c.c. on prime ideals, then D is a Krull domain in which each height-one prime ideal is finitely generated.

In this chapter, all rings are assumed to be commutative and to contain a unity element. A widely used result in commutative ring theory is the Prime Avoidance Theorem, which states that an ideal contained in a finite union of prime ideals is contained in one of the prime ideals. The chapter provides a proposition that gives equivalent conditions on a fixed prime ideal in order that the prime ideal contains the intersection of a family of prime ideals only if the prime ideal contains one of the prime ideals of the family. It also presents basic local results concerning the intersection condition.

Suppose D is an integral domain with quotient field K and that L is an extension field of K. We show in Theorem 4 that if the complete integral closure of D is an intersection of Archimedean valuation domains on K, then the complete integral closure of D in L is an intersection of Archimedean valuation domains on L; this answers a question raised by Gilmer and Heinzer in 1965.

We begin by deening a couple of terms. If F = fF i g i2I is a family of elds, we say that F is the family of residue elds of the ring T if there exists a bijection g : I ! MaxSpec(T) such that F i ' T=g(i) for each i 2 I. Note that the deenition takes into account the multiplicity with which a given eld occurs in F. We say that F is realizable if F is the family of residue elds of a zero-dimensional ring. This paper is concerned with the following question (RF). (RF) What families of elds are realizable? At the outset we can say that while much is known in regard to (RF), the general case of the question remains open, and at this point there is no conjectured answer to (RF). Questions of this type have not received much attention in the study of commutative rings, and it seems worthwhile to ask how (RF) arose. For Heinzer and me, there were two primary sources of motivation. The rst of these was Theorem 12 of I], which states that a ring S is hereditarily zero-dimensional This paper contains material presented in the third 1994 Barrett Lecture, given on April 9. Most of its results come from the paper GH6] (see the list of references for the paper Background and Preliminaries : : : in this volume), and represent work done jointly with W. Heinzer.

Let R=∏α∊A be an infinite product of zero-dimensionalchained rings. It is known that R is either zero-dimensional or infinitedimensional. We prove that a finite-dimensional homo~norphic image of R is of dimension at most one. If each R, is a PIR and if R is infinite-dimensional, then R admits one-dimensional hornomorphic images. However, without the PIR hypothesis on the rings Rα, we present examples to show that R may be infinite-dimensional while each finite-dimensional homomorphic image of R is zero-dimensicnal. JVe prove that a prime ideal of R of positive height is of infinite height, and we give conditions for an infinite product of zero-dimensional local rings to admit a one-dimensional local domain as a honlomorphic image.

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Given a commutative ring R , we investigate the structure of the set of Artinian subrings of R . We also consider the family of zero-dimensional subrings of R . Necessary and sufficient conditions are given in order that every zero-dimensional subring of a ring be Artinian. We also consider closure properties of the set of Artinian subrings of a ring with respect to intersection or finite intersection, and the condition that the set of Artinian subrings of a ring forms a directed family.

Let D D be a Noetherian domain, D ′ D\prime its integral closure, and Int ⁡ ( D ) \operatorname {Int}(D) its ring of integer-valued polynomials in a single variable. It is shown that, if D ′ D\prime has a maximal ideal M ′ M\prime of height one for which D ′ / M ′ D\prime /M\prime is a finite field, then Int ⁡ ( D ) \operatorname {Int}(D) is not Noetherian; indeed, if M ′ M\prime is the only maximal ideal of D ′ D\prime lying over M ′ ∩ D M\prime \cap D , then not even Spec ⁡ ( Int ⁡ ( D ) ) \operatorname {Spec}(\operatorname {Int}(D)) is Noetherian. On the other hand, if every height-one maximal ideal of D ′ D\prime has infinite residue field, then a sufficient condition for Int ⁡ ( D ) \operatorname {Int}(D) to be Noetherian is that the global transform of D D is a finitely generated D D -module.

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 $\leq n$. 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. Necessary and sufficient conditions are given in order that an infinite product of zero-dimensional rings be a directed union of Artinian subrings.