Essential domains and two conjectures in dimension theory

This note investigates two long-standing conjectures on the Krull dimension of integer-valued polynomial rings and of polynomial rings, respectively, in the context of (locally) essential domains.Comment: 7 pages. Proc. Am. Math. Soc. (to appear

On the krull and valuative dimension of D + XDs[X] domains

AbstractIn this paper, we deal with the integral domain D(S,r):=D+(X1,X2,…,Xr)DS[X1, X2,…,Xr], where D is an integral domain and S is a multiplicative set of D. The purpose is to pursue the study, initiated by Costa-Mott-Zafrullah in 1978, concerning the prime ideal structure of such domains. We characterize when D(S,r) is a strong S-domain, a stably strong S-domain, a catenarian domain and a universally catenarian domain. As a consequence, we obtain a new class of non-Noetherian universally catenarian domains. Moreover, we give an explicit formula for the Krull dimension of D(S,r) (depending on S and on the Krull dimensions of D and DS[X1,X2,…,Xr]) and we compute its valuative dimension

On the prime ideal structure of tensor products of algebras

AbstractThis paper is concerned with the prime spectrum of a tensor product of algebras over a field. It seeks necessary and sufficient conditions for such a tensor product to have the S-property, strong S-property, and catenarity. Its main results lead to new examples of stably strong S-rings and universally catenarian rings. The work begins by investigating the minimal prime ideal structure. Throughout, several results on polynomial rings are recovered, and numerous examples are provided to illustrate the scope and sharpness of the results

Trivial Extensions Defined by Coherent-like Conditions

This paper investigates coherent-like conditions and related properties that a trivial extension R := A ∝ E might inherit from the ring A for some classes of modules E. It captures previous results dealing primarily with coherence, and also establishes satisfactory analogues of well-known coherence-like results on pullback constructions. Our results generate new families of examples of rings (with zerodivisors) subject to a given coherent-like condition

The evolving SARS-CoV-2 epidemic in Africa: Insights from rapidly expanding genomic surveillance

INTRODUCTION Investment in Africa over the past year with regard to severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) sequencing has led to a massive increase in the number of sequences, which, to date, exceeds 100,000 sequences generated to track the pandemic on the continent. These sequences have profoundly affected how public health officials in Africa have navigated the COVID-19 pandemic. RATIONALE We demonstrate how the first 100,000 SARS-CoV-2 sequences from Africa have helped monitor the epidemic on the continent, how genomic surveillance expanded over the course of the pandemic, and how we adapted our sequencing methods to deal with an evolving virus. Finally, we also examine how viral lineages have spread across the continent in a phylogeographic framework to gain insights into the underlying temporal and spatial transmission dynamics for several variants of concern (VOCs). RESULTS Our results indicate that the number of countries in Africa that can sequence the virus within their own borders is growing and that this is coupled with a shorter turnaround time from the time of sampling to sequence submission. Ongoing evolution necessitated the continual updating of primer sets, and, as a result, eight primer sets were designed in tandem with viral evolution and used to ensure effective sequencing of the virus. The pandemic unfolded through multiple waves of infection that were each driven by distinct genetic lineages, with B.1-like ancestral strains associated with the first pandemic wave of infections in 2020. Successive waves on the continent were fueled by different VOCs, with Alpha and Beta cocirculating in distinct spatial patterns during the second wave and Delta and Omicron affecting the whole continent during the third and fourth waves, respectively. Phylogeographic reconstruction points toward distinct differences in viral importation and exportation patterns associated with the Alpha, Beta, Delta, and Omicron variants and subvariants, when considering both Africa versus the rest of the world and viral dissemination within the continent. Our epidemiological and phylogenetic inferences therefore underscore the heterogeneous nature of the pandemic on the continent and highlight key insights and challenges, for instance, recognizing the limitations of low testing proportions. We also highlight the early warning capacity that genomic surveillance in Africa has had for the rest of the world with the detection of new lineages and variants, the most recent being the characterization of various Omicron subvariants. CONCLUSION Sustained investment for diagnostics and genomic surveillance in Africa is needed as the virus continues to evolve. This is important not only to help combat SARS-CoV-2 on the continent but also because it can be used as a platform to help address the many emerging and reemerging infectious disease threats in Africa. In particular, capacity building for local sequencing within countries or within the continent should be prioritized because this is generally associated with shorter turnaround times, providing the most benefit to local public health authorities tasked with pandemic response and mitigation and allowing for the fastest reaction to localized outbreaks. These investments are crucial for pandemic preparedness and response and will serve the health of the continent well into the 21st century

Une conjecture sur les anneaux de nagata

RésuméDans un article consacré aux S-domaines forts, Malik et Mott se sont posés la question suivante: étant donné un entier naturel n≥1, a-t-on A[X1,…,Xn] S-domaine fort si et seulement si A(X1,…Xn) S-domaine fort? Nous résolvons la conjecture par la négative. Par ailleurs, après une étude théorique, nous l'améliorons à la fin de cet article

Sur les S-domaines forts de kaplansky

AbstractIn this paper we deal with the study of the stably strong S-domain property for pull-back constructions. Some applications are given in Section 1. The main purpose of Section 2 is to construct a new example of stably strong S-domains

Spectres et theories de la dimension dans les anneaux de polynomes a coefficients dans un produit fibre

SIGLECNRS T Bordereau / INIST-CNRS - Institut de l'Information Scientifique et TechniqueFRFranc