Spectral spaces and ultrafilters

Let $X$ be the prime spectrum of a ring. In [arXiv:0707.1525] the authors define a topology on $X$ by using ultrafilters and they show that this topology is precisely the constructible topology. In this paper we generalize the construction given in [arXiv:0707.1525] and, starting from a set $X$ and a collection of subsets $\mathcal{F}$ of $X$, we define by using ultrafilters a topology on $X$ in which $\mathcal F$ is a collection of clopen sets. We use this construction for giving a new characterization of spectral spaces and several new examples of spectral spaces.Comment: 16 pages. To appear in Communications in Algebr

Pr\"ufer-like conditions on an amalgamated algebra along an ideal

Let $f:A\longrightarrow B$ be a ring homomorphism and let $\mathfrak b$ be an ideal of $B$. In this paper we study Pr\"ufer like conditions in the amalgamation of $A$ with $B$ along $\mathfrak b$, with respect to $f$, a ring construction introduced in 2009 by D'Anna, Finocchiaro and Fontana.Comment: 17 pages. To appear in Houston Journal of Mathematic

Serum creatine kinase isoenzymes in children with osteogenesis imperfecta

This study evaluates serum creatine kinase isoenzyme activity in children with osteogenesis imperfecta to determine its usefulness as a biochemical marker during treatment with bisphosphonate. The changes of creatine kinase (CK) isoenzyme activity during and after discontinuation therapy were observed. These results could be useful in addressing over-treatment risk prevention. Introduction The brain isoenzyme of creatine kinase (CKbb) is highly expressed in mature osteoclasts during osteoclastogenesis, thus plays an important role in bone resorption. We previously identified high serum CKbb levels in 18 children with osteogenesis imperfect (OI) type 1 treated for 1 year with bisphosphonate (neridronate). In the present study, serum CK isoenzymes were evaluated in the same children with continuous versus discontinued neridronate treatment over a further 2-year follow-up period. Methods This study included 18 children with OI type 1, 12 with continued (group A) and 6 with ceased (group B) neridronate treatment. Auxological data, serum biochemical markers of bone metabolism, bone mineral density z-score, and serum total CK and isoenzyme activities were determined in both groups. Results Serum CKbb was progressively and significantly increased in group A (p < 0.004) but rapidly decreased to undetectable levels in group B. In both groups, the cardiac muscle creatine kinase isoenzyme (CKmb) showed a marked decrease, while serum C-terminal telopeptide (CTx) levels were almost unchanged. Conclusions This study provides evidence of the cumulative effect of neridronate administration in increasing serum CKbb levels and the reversible effect after its discontinuation. This approach could be employed for verifying the usefulness of serum CKbb as a biochemical marker in patients receiving prolonged bisphosphonate treatment. Moreover, the decreased serum CKmb levels suggest a systemic effect of these drugs

Topological properties of semigroup primes of a commutative ring

A semigroup prime of a commutative ring $R$ is a prime ideal of the semigroup $(R,\cdot)$. One of the purposes of this paper is to study, from a topological point of view, the space \scal(R) of prime semigroups of $R$. We show that, under a natural topology introduced by B. Olberding in 2010, \scal(R) is a spectral space (after Hochster), spectral extension of \Spec(R), and that the assignment R\mapsto\scal(R) induces a contravariant functor. We then relate -- in the case $R$ is an integral domain -- the topology on \scal(R) with the Zariski topology on the set of overrings of $R$. Furthermore, we investigate the relationship between \scal(R) and the space $\boldsymbol{\mathcal{X}}(R)$ consisting of all nonempty inverse-closed subspaces of \spec(R), which has been introduced and studied in C.A. Finocchiaro, M. Fontana and D. Spirito, "The space of inverse-closed subsets of a spectral space is spectral" (submitted). In this context, we show that \scal( R) is a spectral retract of $\boldsymbol{\mathcal{X}}(R)$ and we characterize when \scal( R) is canonically homeomorphic to $\boldsymbol{\mathcal{X}}(R)$, both in general and when \spec(R) is a Noetherian space. In particular, we obtain that, when $R$ is a B\'ezout domain, \scal( R) is canonically homeomorphic both to $\boldsymbol{\mathcal{X}}(R)$ and to the space \overr(R) of the overrings of $R$ (endowed with the Zariski topology). Finally, we compare the space $\boldsymbol{\mathcal{X}}(R)$ with the space \scal(R(T)) of semigroup primes of the Nagata ring $R(T)$, providing a canonical spectral embedding \xcal(R)\hookrightarrow\scal(R(T)) which makes \xcal(R) a spectral retract of \scal(R(T)).Comment: 21 page

A topological version of Hilbert's Nullstellensatz

We prove that the space of radical ideals of a ring $R$, endowed with the hull-kernel topology, is a spectral space, and that it is canonically homeomorphic to the space of the nonempty Zariski closed subspaces of Spec$(R)$, endowed with a Zariski-like topology.Comment: J. Algebra (to appear

The fallacy of composition and meta-argumentation

Although the fallacy of composition is little studied and trivially illustrated, some view it as ubiquitous and paramount. Furthermore, although definitions regard the concept as unproblematic, it contains three distinct elements, often confused. And although some scholars apparently claim that fallacies are figments of a critic’s imagination, they are really proposing to study fallacies in the context of meta-argumentation. Guided by these ideas, I discuss the important historical example of Michels’s iron law of oligarchy

Meta-Argumentation in Hume’s Critique of the Design Argument

Although Hume’s critique of the design argument is a powerful non-inductive meta-argument, the main line of critical reasoning is not analogical but rather a complex meta-argument. It consists of two parts, one interpretive, the other evaluative. The critical meta-argument advances twelve criticisms: that the design argument is weak because two of its three premises are justified by inadequate subarguments; because its main inference embodies four flaws; and because the conclusion is in itself problematic for four reasons. Such complexity is quite manageable in a meta-argumentation approach

Dialectics, Evaluation, and Argument

Abstract: A critical examination of the dia-lectical approach, focusing on a comparison ofthe illative and the dialectical definitions of argument. I distinguish a moderate, a strong and a hyper dialectical conception of argument. I critique Goldman&apos;s argument for the moderate conception and Johnson&apos;s ar-gument for the strong conception, and argue that the moderate conception is correct. Resume: Un interrogatoire critique sur l&apos;approche dialectical, qui concentre sur une comparaison des definitions d&apos;argument illatives et dialecticales. Je distingue trois conceptions de I &apos; argument: la moden!e, la forte, et la hyper. Je cri-tique l&apos;argument de Goldman qui supporte la conception moderee et celui de Johnson qui support la conception forte, et je pretends que la conception moderee est exact

Economic Reasoning and Fallacy of Composition: Pursuing a Woods-Walton Thesis

Woods and Walton deserve credit for including (in all editions of their textbook Argument) a discussion of “economic reasoning” and its susceptibility to the “fallacy of composition.” Unfortunately, they did not sufficiently pursue the topic, and argumentation scholars have apparently ignored their pioneering effort. Yet, obviously, economic argumentation is extremely important, and economists constantly harp on this fallacy. This paper calls attention to this problem, elaborating my own approach, which is empirical, historical, and meta-argumentational