Robert Gilmer

Some finiteness conditions on the set of overrings of an integral domain

Let D be an integral domain with quotient field K and integral closure D. An overring of D is a subring of K containing D, and O(D) denotes the set of overrings of D. We consider primarily two finiteness conditions on O(D): (FO), which states that O(D) is finite, and (FC), the condition that each chain of distinct elements of O(D) is finite. (FO) is strictly stronger than (FC), but if D = D, each of (FO) and (FC) is equivalent to the condition that D is a Prufer domain with finite prime spectrum. In general D satisfies (FC) iff D satisfies (FC) and all chains of subrings of D containing D have finite length. The corresponding statement for (FO) is also valid.

Products of commutative rings and zero-dimensionality

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

On the complete integral closure of an integral domain

We consider in this paper only commutative rings with identity.When R is considered as a subring of S it will always be assumed that Rand 5 have the same identity. If R is a subring of 5 an element s of 5is said to be integral over R if s is the root of a monic polynomial withcoefficients in R. Following Krull [8], p. 102, we say s is almost integralover R provided all powers of s belong to a finite * i?-submodule of S.If R

Multiplication rings as rings in which ideals with prime radical are primary

A commutative ring R is called an AM-ring (for allgemeine multiplikationsring) if whenever A and B are ideals of R with A properly contained in B, then there is an ideal C of R such that A = BC. An AM-ring R in which RA = A for each ideal A of R is called a multiplication ring. Krull introduced the notion of a multiplication ring in [11], [13]. Akizuki is responsible for the more general concept of an AM-ring in [1], but Mori has developed most of the structure theory for such rings in [14], [15], [16], [17], and [18]. An important property of an AM-ring R is that R satisfies what Gilmer called condition (*) in [ 7 ] and [8 ]: An ideal of R with prime radical is primary. In ? 1, new results concerning rings in which (*) holds are given. These are applied to obtain structure theorems for AM-rings in ? 2. In [10, p. 737], Krull introduced the notion of the kernel of an ideal A in a commutative ring R, which is defined thusly: if Pa } is the collection of minimal prime ideals of A, then by an isolated primary component of A belonging to Pa we mean the intersection Qa of all Pay-primary ideals which contain A. The kernel of A is the intersection of all Qas. Mori considered rings in which every ideal is equal to its kernel in [ 16] and [ 17 ]. In ? 1, it is shown that a ring R satisfies (*) if and only if every ideal of R is equal to its kernel. In ?2 of this paper, we consider rings R satisfying condition (F): If A and P are ideals of R such that P is prime and A is properly contained in P, there is an ideal B such that A = PB. Theorems 12 and 13 show that such a ring is an AM-ring. This generalizes a result proved by Mott in [19] for rings with unit. This theorem might be compared with the result of Cohen [3, Theorem 2, p. 29], that a ring in which every prime ideal is finitely generated is a Noetherian ring and to the theorem of Nakano in [ 20, p. 234] which states that every nonzero ideal of an integral domain D with unit is invertible provided every nonzero prime ideal of D is invertible. In addition, several new sets of necessary and sufficient conditions that a ring R be an AM-ring are given in ? 2. An equally significant aspect of ? 2 in our eyes is that in the process of proving Theorem 12, many of the known results concerning AM-rings are proved in a way we feel is clearer and more straight-

The Laskerian property, power series rings and Noetherian spectra

Let R be a commutative ring with identity. An ideal Q of R is primary if each zero divisor of the ring R/ Q is nilpotent, and Q is strongly primary if Q is primary and contains a power of its radical. In the terminology of Bourbaki [B, Ch. IV, pp. 295, 298], the ring R is Laskerian if each ideal of R is a finite intersection of primary ideals, and R is strongly Laskerian if each ideal of R is a finite intersection of strongly primary ideals. It is well known that

Ideals contracted from a Noetherian extension ring

Abstract Let R be a commutative ring with identity such that for each ideal A of R, there exists a Noetherian unitary extension ring T(A) of R such that A is contracted from T(A). We investigate the structure of R. The context in which this topic has usually been considered is where R is an integal domain and T(A) is an overring of R. Under these hypotheses we show that R is Neotherian if R is one-dimensional. In the general case, R is strongly Laskerian, has Noetherian spectrum, and satisfies certain chain conditions for quotient ideals, but R need not be Noetherian.

On the Number of Generators of an Invertible Ideal

Prtifer domains, and by means of this construction we give an example which (a) shows that none of the sufficient conditions given in Sections 1 and 2 in order that an invertible ideal have a basis of two elements are necessary, and (b) answers in the negative a question raised by Matlis in [15, p. 1511. All rings considered in this paper are assumed to be commutative. 1.

On the contents of polynomials

In [13, p. 24], Prufer establishes the following result. (See also [2, Exercise 21, p. 97].) Let J be an integral domain with identity having quotient field K and letf, g, h:K [X] be such that h =fg. If A, B, and C denote the fractional ideals of J generated by the coefficients of f, g, and h, respectively, and if n is the degree of the polynomial g, then A C = A +B. This result is essentially what Krull in [7, p. 128] calls the Hilfssatz von Dedekind-Mertens, although the results which Dedekind in [4] and Mertens in [II ] prove are not so general as Prufers theorem. In this note, we generalize the theorem cited above, first to the case where J is an arbitrary subring of the commutative ring K, and then to the case of polynomials in finitely many indeterminates. We conclude with some applications of the results obtained. We use consistently the following notation in this paper. R denotes a subring of a commutative ring S. If f is a polynominal over S, Yf denotes the set of coefficients of f, and Af denotes the R-submodule of S generated by Yf; we call Af the R-content of f. The following result is straightforward, but we list it here for the sake of reference.

ZERO-DIMENSIONALITY IN COMMUTATIVE RINGS

If {Ra}aeA is a family of zero-dimensional subrings of a commu- tative ring T , we show that f)aeA Rc is also zero-dimensional. Thus, if R is a subring of a zero-dimensional subring * T (a condition that is satisfied if and only if a power of rT is idempotent for each r 6 R), then there exists a unique minimal zero-dimensional subring R° of T containing R. We inves- tigate properties of R° as an A-algebra, and we show that R° is unique, up to /{-isomorphism, only if R itself is zero-dimensional.

Artinian subrings of a commutative ring

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