The Krull dimension of power series rings over non-SFT rings

AbstractLet R be a commutative ring with identity. We show that the Krull dimension of the power series ring R〚X〛 can be uncountably infinite, i.e., there exists an uncountably infinite chain of prime ideals in R〚X〛, even if dimR is finite. In fact, we show that dimR〚X〛 is uncountably infinite if R is a non-SFT ring, which is an improvement of Arnold’s result

Integral closure of rings of integer-valued polynomials on algebras

Let $D$ be an integrally closed domain with quotient field $K$. Let $A$ be a torsion-free $D$-algebra that is finitely generated as a $D$-module. For every $a$ in $A$ we consider its minimal polynomial $\mu_a(X)\in D[X]$, i.e. the monic polynomial of least degree such that $\mu_a(a)=0$. The ring ${\rm Int}_K(A)$ consists of polynomials in $K[X]$ that send elements of $A$ back to $A$ under evaluation. If $D$ has finite residue rings, we show that the integral closure of ${\rm Int}_K(A)$ is the ring of polynomials in $K[X]$ which map the roots in an algebraic closure of $K$ of all the $\mu_a(X)$, $a\in A$, into elements that are integral over $D$. The result is obtained by identifying $A$ with a $D$-subalgebra of the matrix algebra $M_n(K)$ for some $n$ and then considering polynomials which map a matrix to a matrix integral over $D$. We also obtain information about polynomially dense subsets of these rings of polynomials.Comment: Keywords: Integer-valued polynomial, matrix, triangular matrix, integral closure, pullback, polynomially dense set. accepted for publication in the volume "Commutative rings, integer-valued polynomials and polynomial functions", M. Fontana, S. Frisch and S. Glaz (editors), Springer 201

Some closure operations in Zariski-Riemann spaces of valuation domains: a survey

In this survey we present several results concerning various topologies that were introduced in recent years on spaces of valuation domains

Comparative genomics of plant-asssociated Pseudomonas spp.: Insights into diversity and inheritance of traits involved in multitrophic interactions

We provide here a comparative genome analysis of ten strains within the Pseudomonas fluorescens group including seven new genomic sequences. These strains exhibit a diverse spectrum of traits involved in biological control and other multitrophic interactions with plants, microbes, and insects. Multilocus sequence analysis placed the strains in three sub-clades, which was reinforced by high levels of synteny, size of core genomes, and relatedness of orthologous genes between strains within a sub-clade. The heterogeneity of the P. fluorescens group was reflected in the large size of its pan-genome, which makes up approximately 54% of the pan-genome of the genus as a whole, and a core genome representing only 45–52% of the genome of any individual strain. We discovered genes for traits that were not known previously in the strains, including genes for the biosynthesis of the siderophores achromobactin and pseudomonine and the antibiotic 2-hexyl-5-propyl-alkylresorcinol; novel bacteriocins; type II, III, and VI secretion systems; and insect toxins. Certain gene clusters, such as those for two type III secretion systems, are present only in specific sub-clades, suggesting vertical inheritance. Almost all of the genes associated with multitrophic interactions map to genomic regions present in only a subset of the strains or unique to a specific strain. To explore the evolutionary origin of these genes, we mapped their distributions relative to the locations of mobile genetic elements and repetitive extragenic palindromic (REP) elements in each genome. The mobile genetic elements and many strain-specific genes fall into regions devoid of REP elements (i.e., REP deserts) and regions displaying atypical tri-nucleotide composition, possibly indicating relatively recent acquisition of these loci. Collectively, the results of this study highlight the enormous heterogeneity of the P. fluorescens group and the importance of the variable genome in tailoring individual strains to their specific lifestyles and functional repertoir

The Rsm regulon of plant growth-promoting Pseudomonas fluorescens SS101: role of small RNAs in regulation of lipopeptide biosynthesis

The rhizobacterium Pseudomonas fluorescens SS101 inhibits growth of oomycete and fungal pathogens, and induces resistance in plants against pathogens and insects. To unravel regulatory pathways of secondary metabolite production in SS101, we conducted a genome-wide search for sRNAs and performed transcriptomic analyses to identify genes associated with the Rsm (repressor of secondary metabolites) regulon. In silico analysis led to the identification of 16 putative sRNAs in the SS101 genome. In frame deletion of the sRNAs rsmY and rsmZ showed that the Rsm system regulates the biosynthesis of the lipopeptide massetolide A and involves the two repressor proteins RsmA and RsmE, with the LuxR-type transcriptional regulator MassAR as their most likely target. Transcriptome analyses of the rsmYZ mutant further revealed that genes associated with iron acquisition, motility and chemotaxis were significantly upregulated, whereas genes of the type VI secretion system were downregulated. Comparative transcriptomic analyses showed that most, but not all, of the genes controlled by RsmY/RsmZ are also controlled by the GacS/GacA two-component system. We conclude that the Rsm regulon of P.¿fluorescens SS101 plays a critical role in the regulation of lipopeptide biosynthesis and controls the expression of other genes involved in motility, competition and survival in the plant rhizosphere