On the Strong (A)-Rings of Mahdou and Hassani

A (commutative unital) ring R with only finitely many minimal prime ideals (for instance, a Noetherian ring) is reduced and a strong (A)-ring if and only if R is an integral domain. Thus, the smallest reduced ring which has Property A but is not a strong (A)-ring is ${\mathbb{Z}_{2} \times \mathbb{Z}_{2}}$ . A Noetherian ring R is a strong (A)-ring if and only if Ass _R (R) has a unique maximal element.     

Monofluorinated di- and tetrahydropyrans via Prins-type cyclisations

The synthesis of a range of fluorinated heterocycles is described via a Lewis acid-mediated Prins-type cyclisation.     

The Use of Laser Backscatter Instrumentation for the on-line measurement of the particle size distribution of emulsions

The response of the Partec laser backscatter particle size analyzer has been tested under a variety of conditions. Various materials posessing different light scattering characteristics have been examined and the response of the Partec instrument has been evaluated. In particular the system response to transparent particle systems (e.g. oil in water emulsions) tas been studied. It has been found that the Partec laser backscatter instrument provides reasonably accurate and reproducible results when used to characterize opaque, highly reflective particles such as hydrated alumina, but suffers from a lack of sensitivity to fine transparent particles such as oil droplets. The instrument does, however, readily detect the presence of opaque particle contaminants in oil-water emulsions.     

A going-up theorem for arbitrary chains of prime ideals

We prove that if an extension R ? T of commutative rings satisfies the going-up property (for instance, if T is an integral extension of R), then any increasing chain of prime ideals of R (indexed by an arbitrary linearly ordered set) is covered by some corresponding chain of prime ideals of T. As a corollary, we recover the recent result of Kang and Oh that any such chain of prime ideals of an integral domain D is covered by a corresponding chain in some valuation overring of D.     

Lifting trees of prime ideals to bezout extension domains

We prove that if an extension R ? T of commutative rings satisfies the going-up property, then any tree of prime ideals of R with at most two branches or in which each branch has finite length is covered by some corresponding tree of prime ideals of T. In particular, if R ? T is an integral extension and R is Noetherian, then each tree in Spec(R) can be covered by a tree in Spec(T). We also prove that if R is an integral domain, then each tree T in Spec(/2) can be covered by a tree in Spec(T) for some Bezout domain T containing R. If T has only finitely many branches, it can further be arranged that the Bezout domain T be an overring of R. However, in general, it cannot be arranged that T be covered from a Prüfer overring of R, thus answering negatively a question of D D. Anderson.     

Transfer Results for the FIP and FCP Properties of Ring Extensions

For an extension E: R ? S of (commutative) rings and the induced extension F: R(X) ? S(X) of Nagata rings, the transfer of the FCP and FIP properties between E and F is studied. Then F has FCP ? E has FCP. The extensions E for which F has FIP are characterized. While E has FIP whenever F has FIP, the converse fails for certain subintegral extensions; it does hold if E is integrally closed, seminormal, or subintegral with R quasi-local having infinite residue field. If F has FIP, conditions are given for the sets of intermediate rings of E and F to be order-isomorphic.     

On characterizations of the ordered field of real numbers

This self-contained note could find classroom use in an introductory course on analysis. It is proved that an ordered field F is complete (that is, order-isomorphic to the field of real numbers) if and only if each bounded monotonic sequence in F converges in F. Also established is the key tool that an ordered field is complete if and only if it is Archimedean and Cauchy-complete, along with a number of characterizations of Archimedean fields.