Joe L. Mott

The construction D + XDS[X]

If D is a commutative integral domain and S is a multiplicative system in D, then Tfs) = D + XD,[X] is the subring of the polynomial ring D,[X] con- sisting of those polynomials with constant term in D. In the special case where S = D* = D\(O), we omit the superscript and let T denote the ring D + XK[XJ, where K is the quotient field of D. Since Tfs) is the direct limit of the rings D[X/s], where s E S, we can conclude that many properties hold in T@) because these properties are preserved by taking polynomial ring extensions and direct limits. Moreover, the ring Tcs) is the symmetric algebra S,(D,) of D, considered as a D-module. In addition, Ds[Xj is a quotient ring of Tts) with respect to S; in fact, in the terminology of [lo], Tfs) is the composite of D and D,[iYj over the ideal XDJX]. (The most familiar of the composite constructions is the so-called D + M construction [l], where generally M is the maximal ideal of a valuation ring.) The ring T ts), therefore, provides a test case for many questions about direct limits, symmetric algebras, and composites. The state of our knowledge of T is considerably more advanced than that of VJ; generally speaking, we often show that a property holds in T if and only if it holds in D. In other cases we show that Tcs) does not have a given property if D, # K. For example, if T(S) is a Priifer domain, then D,[xJ is a Prtifer domain and D, is therefore equal to K. We show that T is Priifer (Bezout) if and only if D is Prtifer (Bezout). Yet Tts) is a GCD-domain if D is a GCD- domain and the greatest common divisor of d and X exists in

Cohen-Kaplansky domains: Integral domains with a finite number of irreducible elements

We define an integral domain R to be a Cohen-Kaplansky domain (CK domain) if every element of R is a finite product of irreducible elements and R has only finitely many nonassociate irreducible elements. The purpose of this paper is to investigate CK domains. Many conditions equivalent to R being a CK domain are given, for example, R is a CK domain if and only if R is a one-dimensional semi-local domain and for each nonprincipal maximal ideal M of R, RM is finite and RM is analytically irreducible, or, if and only if G(R), the group of divisibility of R, is finitely generated and rank G(R)= ¦Max(R)¦. We show that a CK domain is a certain special type of composite or pullback of a subring of a finite homomorphic image of a semilocal PID. Noetherian domains with G(R) finitely generated are also investigated.

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-

An axiomatic approach to fuzzy set theory

Abstract An axiomatic basis for the concept of “fuzzy set” is established, following the classical class theory developed by von Neumann and Bernays (VNB). The set of axioms we exhibit is the minimal set that enables a formal definition of fuzzy sets, without assuming a universe of discourse, and requires fewer axioms than any other published formal definition of fuzzy sets.

On Krull domains

One aim of this article is to provide for Krull domains a star-operation analogue of the following result: An integral domain D is a Dedekind domain if and only if each nonzero ideal A of D is strongly two generated. A nonzero ideal A of an integral domain D is called strongly two generated if for each x e A\{0} there is y e A such that A = xD + yD. Lantz and Martin show in [17] that a strongly two generated ideal is invertible. Following this lead we define a strongly *-type 2 ideal, for a star-operation *, as a nonzero ideal A such that for each x e A\{0}, there is y ~ A* such that (x, y)* = A*. Then in Section I we characterize Krult domains in terms of strongly *-type 2 ideals. Recently there has been considerable activity [1, 7, 12, 26] (some of it inspired by an earlier preprint version of the present paper) in characterizing a Krult domain in terms of the ,-invertibility of some or all fractional ideals of D. These results are interesting in that they indicate that most of the characterizations of Dedekind domains have *-oper- ation analogues for Krull domains. In Section 2 we continue this line of investigation by coordinating some of the recent results with some new characterizations of Krull domains in terms of *-invertibility.

Factoriality in Riesz groups

Abstract Throughout let G = (G,+,≤, 0) denote a Riesz group, where + is not necessarily a commutative operation. Call x ∈ G homogeneous if x > 0 and for all h, k ∈ (0, x] there is t ∈ (0, x] such that t ≤ h, k. In this paper we develop a theory of factoriality in Riesz groups based on the fact that if x ≤ G and x is a finite sum of homogeneous elements then x is uniquely expressible as a sum of finitely many mutually disjoint homogeneous elements. We then compare our work with existing results in lattice-ordered groups and in (commutative) integral domains.

Groups of Divisibility: A Unifying Concept for Integral Domains and Partially Ordered Groups

The theory of divisibility and factorization in an integral domain is essentially the study of a partially ordered abelian group. This is because if D is an integral domain with identity, K a field containing D, and K* the set of nonzero elements of K, then the divisibility relation (a divides b for a,b∈K* if and only if b/a∈D) determines a partial order on K* modulo the units of D. To be sure, the divisibility relation is reflexive and transitive on K* and so determines a preordering compatible with the group structure of K* (if a divides b, then ac divides bc for all c∈K*).

Integrally closed subrings of an integral domain

Let D be an integral domain with identity having quotient field K. This paper gives necessary and sufficient conditions on D in order that each integrally closed subring of D should belong to some subclass of the class of integrally closed domains; some of the subclasses considered are the completely integrally closed domains, Prufer domains, and Dedekind domains. 1. The class of integrally closed domains contains several classes of domains which are of fundamental importance in commutative algebra. Unique factorization domains, Krull domains, domains of finite character, Priifer domains, completely integrally closed domains, Dedekind domains, and principal ideal domains are examples of such subclasses of the class of integrally closed domains. This paper considers the problems of determining, conversely, necessary and sufficient conditions on an integral domain with identity in order that each of its integrally closed subrings should belong to some subclass of the class of integrally closed domains. An example of a typical result might be Theorem 2.3: If J is an integral domain with identity having quotient field K, then conditions (1) and (2) are equivalent. (1) Each integrally closed subring of J is completely integrally closed. (2) Either J has characteristic 0 and K is algebraic over the field of rational numbers or J has characteristic p # 0 and K has transcendence degree at most one over its prime subfield. If J is integrally closed, then conditions (1) and (2) are equivalent to: (3) Each integrally closed subring of J with quotient field K is completely integrally closed. In considering characterizations of integral domains with identity for which every integrally closed subring is Dedekind or almost Dedekind (?3), we are led to use some results of W. Krull to prove Theorem 4.1, which establishes the existence of, as well as a method for constructing, a field with certain specified valuations. We then use this theorem to construct an example of an infinite separable algebraic extension field K of FLp(X) such that the integral closure J of FLp[X] in K Received by the editors April 7, 1970. AMS 1969 subject classifications. Primary 1315, 1350; Secondary 1320.

The GCD property and irreduciable quadratic polynomials

The proof of the following theorem is presented: If D is, respectively, a Krull domain, a Dedekind domain, or a Prufer domain, then D is correspondingly a UFD, a PID, or a Bezout domain if and only if every irreducible quadratic polynomial in D[X] is a prime element.

An algebraic proof of a theorem of A. Robinson

A. Robinson has used mathematical logic to obtain a theorem concerning systems of polynomial equations with only finitely many solutions; this paper contains an algebraic proof of Robinsons theorem, based primarily on various equivalent forms of Huberts Nullstellensatz. In [4, p. 38], Robinson uses principles of logic to solve some problems in algebra. By use of lower predicate calculus and Gödels completeness theorems [4, p. 12], he gives elegant proofs of some known results and at least one new result; Robinson indicates that this new result has no readily accessible purely mathematical solution independent of his approach. Statement of the Theorem. Let fi, ■ ■ ■ , ft be polynomials in « indeterminates with integer coefficients (that is, /i, • ■ • , ft ÇzZ[Xi, • ■ ■ , Xn]) and denote byff the polynomial which is obtained from fi by reducing its coefficients modulo p. If the system of equations fi = 0,/2 = 0, • • • ,ft = 0 has at most m solutions in any extension field of the rationals, then there is a prime q such that for each prime p>q, the system ff = 0, fl = 0, • • • ,/? = 0 has at most m solutions in any extension field of the finite field Zv. While we concede the power and beauty of Robinsons approach, the purpose of this note is to give an elementary algebraic proof of the preceding theorem. Our proof will consist of three parts; the first two parts will be separated into lemmas. First we remark that the Hubert Nullstellensatz, as commonly stated [8, p. 164], asserts that if a polynomial f(X\, ■ ■ ■ , X„) over a field k vanishes at every zero of an ideal A of k[Xi, • ■ ■ , Xn], then some power of / is in A. On the basis of this result (or by use of Zariskis Nullstellensatz [7, pp. 362-363]), one can establish a oneto-one correspondence between the set of maximal ideals of k[Xu ■ ■ ■ , X„] and a certain set of equivalence classes in »-dimenReceived by the editors October 8, 1970. AMS 1970 subject classifications. Primary 14A10; Secondary 13L05.