Andrew J. Hetzel

On chain morphisms of commutative rings

RésuméOn dit qu’un homomorphismef :A →B d’anneaux commutatifs est un morphisme de chaîne (resp., den-chaîne pour un entiern ≥ 1) si toute chaîne d’idéaux premiers (resp., d’au plusn idéaux premiers) deA se relève en une chaîne d’idéaux premiers deB. Sif est un morphisme den-chaîne, alorsf n’est pas forcément un morphisme de (n + 1)-chaîne, même siA etB sont des anneaux intègres, doncf n’est pas un morphisme de chaîne. Sif est un morphisme den-chaîne pour toutn, alorsf est un morphisme de chaîne. Un morphisme de chaîne n’est pas forcément un morphisme de chaîne universel. Pour tout entiern ≥ 2,f est universellement un morphisme den-chaîne si et seulement sif est universellement un morphisme de chaîne. Un morphisme qui est universellement de chaîne et universellement incomparable n’est pas nécessairement entier, même siA etB sont des anneaux intègres de dimension 1 (au sens de Krull).

On the Relationship between Zero-sums and Zero-divisors of Semirings

Abstract. In this article, we generalize a well-known result of Hebisch and Weinert thatstates that a finite semidomain is either zerosumfree or a ring. Specifically, we show thatthe class of commutative semirings S such that S has nonzero characteristic and everyzero-divisor of S is nilpotent can be partitioned into zerosumfree semirings and rings. Inaddition, we demonstrate that if S is a finite commutative semiring such that the set ofzero-divisors of S forms a subtractive ideal of S, then either every zero-sum of S is nilpo-tent or S must be a ring. An example is given to establish the existence of semirings in thislatter category with both nontrivial zero-sums and zero-divisors that are not nilpotent. 1. IntroductionThis article is devoted to an exploration of how ideal-theoretic considerations incommutative semirings, particularly finite commutative semirings, impact the mul-tiplicative behavior of those elements of the semiring that have additive inversesin the semiring. The general question as to the algebraic nature of these so-called“zero-sums” of a semiring is one of the most central in the theory of semirings. Weare especially motivated by a result of Hebisch and Weinert [9, Corollary 3.4, p. 81]that establishes that the class of finite semidomains can be partitioned by the an-tipodal properties of being zerosumfree (that is, only the zero element is a zero-sumof the semiring) and being a ring (where, by definition, every element is a zero-sumof the semiring). Of course, there exist infinite semidomains with nontrivial zero-sums that are not rings; for example, the polynomial semiring XZ[X] + N, where

On the Ascent of Properties Related to Unique Factorization Domains

In this article, we develop equivalent conditions for a certain class of monoidal transform to inherit either the property of being a completely integrally closed domain that satisfies the ascending chain condition on principal ideals, the property of being a Mori domain, the property of being a Krull domain, or the property of being a unique factorization domain, respectively. Such a class of monoidal transform is given in terms of an (analytically) independent set that forms a prime ideal in the base domain. Characterizations are provided illustrating the necessity of the “prime ideal” hypothesis when the base domain is a Noetherian unique factorization domain.

On the Ascent of Accp to Simple Overrings

ABSTRACT In this article, we raise the question as to what conditions permit a simple overring of a domain R —that is, a domain of the form for some f, g ∈ R such that g ≠ 0—to inherit the ascending chain condition on principal ideals from R . Our main theorem reveals that, if g is a prime element of R , the complete answer can be found by considering the Archimedean property. We then, in turn, use this theorem to establish equivalent conditions for a certain class of simple overring to inherit the property of being a unique factorization domain from R .

A class of practically nowhere differentiable functions on the complex plane

This note could find classroom use in an introductory course on complex analysis. Using some of the most significant theorems from complex analysis, the main result provides a simple method for transforming many elementary functions (defined on the complex plane) into everywhere continuous functions that are differentiable only on a nowhere dense set. Accordingly, such continuous functions are termed ‘practically nowhere differentiable’. The twofold pedagogical value of this method is that (1) students can readily generate examples of everywhere continuous, practically nowhere differentiable functions that do not require any direct appeal to infinite series, and (2) the often dynamical difference between the behaviour of functions of a complex variable and functions of a real variable is showcased.

Classroom note: Ahmes expansions of rational numbers of length two

This note could find classroom use in a first course on number theory. Three algebraic characterizations of the positive rational numbers q for which there exist positive integers m 1<m 2 such that q=1/m 1+1/m 2 are provided. These characterizations lead to determination of the minimum ‘length’ k for which a given rational number q of the form 2/n or 3/n, with n>1, can be expressed as 1/m 1 +…+1/mk , for positive integers m1 < … < mk . A partial result for the numbers 4/n is also given. In a probabilistic vein, it is shown, for each k≥1, that as →∞, 0 is the limit of the proportion of positive proper fractions a/n in lowest terms admitting an expansion 1/m 1 +…+1/mk of the above type.