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Cédric Milliet - One of the best experts on this subject based on the ideXlab platform.

  • $\rm NIP$, and ${\rm NTP}_2$ Division Rings of prime characteristic
    2019
    Co-Authors: Cédric Milliet
    Abstract:

    Combining a characterisation by Bélair, Kaplan, Scanlon and Wagner of certain $\rm NIP$ valued fields of characteristic $p$ with Dickson's construction of cyclic algebras, we provide examples of noncommutative $\rm NIP$ Division Ring of characteristic $p$ and show that an $\rm NIP$ Division Ring of characteristic $p$ has finite dimension over its centre, in the spirit of Kaplan and Scanlon's proof that infinite $\rm NIP$ fields have no Artin-Schreier extension. The result extends to ${\rm NTP}_2$ Division Rings of characteristic $p$, using results of Chernikov, Kaplan and Simon. We also highlight consequences of our proofs that concern $\rm NIP$ or simple difference fields.

  • $\rm NIP$, and ${\rm NTP}_2$ Division Rings of prime characteristic
    arXiv: Logic, 2019
    Co-Authors: Cédric Milliet
    Abstract:

    Combining a characterisation by Belair, Kaplan, Scanlon and Wagner of certain $\rm NIP$ valued fields of characteristic $p$ with Dickson's construction of cyclic algebras, we provide examples of noncommutative $\rm NIP$ Division Ring of characteristic $p$ and show that an $\rm NIP$ Division Ring of characteristic $p$ has finite dimension over its centre, in the spirit of Kaplan and Scanlon's proof that infinite $\rm NIP$ fields have no Artin-Schreier extension. The result extends to ${\rm NTP}_2$ Division Rings of characteristic $p$, using results of Chernikov, Kaplan and Simon. We also highlight consequences of our proofs that concern $\rm NIP$ or simple difference fields.

  • Linear Algebra over a Division Ring
    2016
    Co-Authors: Cédric Milliet
    Abstract:

    We consider an analogue of the Zariski topology over a Division Ring~$\R$ equipped with a Ring morphism $\sigma:\R\rightarrow\R$. A basic closed subset of $\R^n$ is given by the zero set of a (finite) family of linear combinations of $\left\{\sigma^{i_1}(x_1),\dots,\sigma^{i_n}(x_n):(i_1,\dots,i_n)\in\mathbb N^n\right\}$ having left coefficients in~$\R$. This enables us to define elementary notions of algebraic geometry: algebraic sets, $\sigma$-morphisms and comorphisms, a notion of Zariski dimension, a notion of radical component of an algebraic set. We classify the algebraic sets over $\R$ up to $\sigma$-isomorphisms when $\sigma$ is onto $\R$ and $[\R:{\rm Fix}(\sigma)]$ infinite (and as a by-product, the additive algebraic groups over a perfect field), and show that any Division Ring with infinite $[\R:{\rm Fix}(\sigma)]$ has an extension in which each affine polynomial $r+r_0x+r_1\sigma(x)+\dots+r_n\sigma^n(x)$ has a root. In such an extension, Chevalley's projection Theorem for constructible sets holds, as well as affine Nullstellensatze. These results are intended to be applied in a further paper to Division Rings that do not have Shelah's independence property.

  • Fields with few types
    The Journal of Symbolic Logic, 2013
    Co-Authors: Cédric Milliet
    Abstract:

    Let R be an associative Ring with possible extra structure. R is said to be weakly small if there are countably many 1-types over any finite subset of R. It is locally P if the algebraic closure of any finite subset of R has property P. It is shown here that a field extension of finite degree of a weakly small field either is a finite field or has no Artin-Schreier extension. A weakly small field of characteristic 2 is finite or algebraically closed. Every weakly small Division Ring of positive characteristic is locally finite dimensional over its centre. The Jacobson radical of a weakly small Ring is locally nilpotent. Every weakly small Division Ring is locally, modulo its Jacobson radical, isomorphic to a product of finitely many matrix Rings over Division Rings.

  • Stable Division Rings
    The Journal of Symbolic Logic, 2011
    Co-Authors: Cédric Milliet
    Abstract:

    AbstractIt is shown that a stable Division Ring with positive characteristic has finite dimension over its centre. This is then extended to simple Division Rings.

Jairo Z Goncalves - One of the best experts on this subject based on the ideXlab platform.

Bui Xuan Hai - One of the best experts on this subject based on the ideXlab platform.

  • On Division subRings normalized by almost subnormal subgroups in Division Rings
    Periodica Mathematica Hungarica, 2019
    Co-Authors: Trinh Thanh Deo, Mai Hoang Bien, Bui Xuan Hai
    Abstract:

    Let D be a Division Ring with infinite center, K a proper Division subRing of D and N an almost subnormal subgroup of the multiplicative group \(D^*\) of D. The aim of this paper is to show that if K is N-invariant and N is non-central, then K is central. Some examples of almost subnormal subgroups in Division Rings that are not subnormal are also given.

  • On multiplicative subgroups in Division Rings.
    Journal of Algebra and Its Applications, 2016
    Co-Authors: Bui Xuan Hai
    Abstract:

    Let $D$ be a Division Ring. In this paper, we investigate properties of subgroups of an arbitrary subnormal subgroup of the multiplicative group $D^*$ of $D$. The new obtained results generalize some previous results on subgroups of $D^*$.

  • On the Gelfand-Kirillov dimension of weakly locally finite Division Rings
    arXiv: Rings and Algebras, 2015
    Co-Authors: Bui Xuan Hai, Mai Hoang Bien, Trinh Thanh Deo
    Abstract:

    Weakly locally finite Division Rings were considered in \cite{dbh}. In this paper, firstly, we prove that weakly locally finite Division Rings are exactly locally PI Division Rings, and then we consider the Gelfand-Kirillov dimension of such Division Rings. It is shown that the GKdim of any weakly locally finite Division Ring is either non-negative integer or infinite. Moreover, for any integer $n\geq 0$ or $n=\infty$, we construct a weakly locally finite Division Ring whose GKdim is $n$. Further, we investigate some questions related with the Kurosh Problem for Division Rings. In particular, we give a positive answer to the Kurosh Problem for weakly locally finite Division Rings. Finally, one of Herstein's conjectures is also investigated.

  • A note on the existence of non-cyclic free subgroups in Division Rings
    Archiv der Mathematik, 2013
    Co-Authors: Bui Xuan Hai, Nguyen Kim Ngoc
    Abstract:

    A Division Ring D is said to be weakly locally finite if for every finite subset \({S \subset D}\), the Division subRing of D generated by S is centrally finite. It is known that the class of weakly locally finite Division Rings strictly contains the class of locally finite Division Rings. In this note we prove that every non-central subnormal subgroup of the multiplicative group of a weakly locally finite Division Ring contains a non-cyclic free subgroup. This generalizes the previous result by Goncalves for centrally finite Division Rings.

  • On subgroups in Division Rings of type $2$
    Studia Scientiarum Mathematicarum Hungarica, 2012
    Co-Authors: Bui Xuan Hai, Trinh Thanh Deo, Mai Hoang Bien
    Abstract:

    Let $D$ be a Division Ring with center $F$. We say that $D$ is a {\em Division Ring of type $2$} if for every two elements $x, y\in D,$ the Division subRing $F(x, y)$ is a finite dimensional vector space over $F$. In this paper we investigate multiplicative subgroups in such a Ring.

Miguel Vicente - One of the best experts on this subject based on the ideXlab platform.

  • role of escherichia coli ftsn protein in the assembly and stability of the cell Division Ring
    Molecular Microbiology, 2010
    Co-Authors: Ana Isabel Rico, Marta Garciaovalle, Pilar Palacios, Mercedes Casanova, Miguel Vicente
    Abstract:

    Deprivation of FtsN, the last protein in the hierarchy of divisome assembly, causes the disassembly of other elements from the Division Ring, even extending to already assembled proto-Ring proteins. Therefore the stability and function of the divisome to produce Rings active in septation is not guaranteed until FtsN is recruited. Disassembly follows an inverse sequential pathway relative to assembly. In the absence of FtsN, the frequencies of FtsN and FtsQ Rings are affected similarly. Among the proto-Ring components, ZipA are more sensitive than FtsZ or FtsA Rings. In contrast, removal of FtsZ leads to an almost simultaneous disappearance of the other elements from Rings. Although restoration of FtsN allows for a quick reincorporation of ZipA into proto-Rings, the de novo joint assembly of the three components when FtsZ levels are restored to FtsZ-deprived filaments is even faster. This suggests that the recruitment of ZipA into FtsZ-FtsA incomplete proto-Rings may require first a period for the reversal of these partial assemblies.

  • the order of the Ring assembly of escherichia coli cell Division components
    Molecular Microbiology, 2006
    Co-Authors: Miguel Vicente, Ana Isabel Rico
    Abstract:

    Topological cues appear to override temporal events in the assembly of the Escherichia coli cell Division Ring. When a procedure that allows the recruitment of Ring components based on their topological properties is used, a concerted mode of assembly of several components of the divisome, rather than a strict linear mode, is revealed. Three multimolecular complexes, the proto-Ring, the periplasmic connector and the peptidoglycan factory, show some degree of concertation for their assembly. In addition, back-recruitment of all late proteins except FtsN into the Division Ring occurs even in the absence of proteins incorporated at earlier stages, i.e. FtsA or FtsQ.

  • role of two essential domains of escherichia coli ftsa in localization and progression of the Division Ring
    Molecular Microbiology, 2004
    Co-Authors: Ana Isabel Rico, Jesus Mingorance, Marta Garciaovalle, Miguel Vicente
    Abstract:

    The FtsA protein is a member of the actin superfamily that localizes to the bacterial septal Ring duRing cell Division. Deletions of domain 1C or the S12 and S13 beta-strands in domain 2B of the Escherichia coli FtsA, previously postulated to be involved in dimerization, result in partially active proteins that do not allow the normal progression of septation. The truncated FtsA protein lacking domain 1C (FtsADelta1C) localizes in correctly placed Division Rings, together with FtsZ and ZipA, but does not interact with other FtsA molecules in the yeast two-hybrid assay, and fails to recruit FtsQ and FtsN into the Division Ring. The Rings containing FtsADelta1C are therefore incomplete and do not support Division. The production of high levels of FtsADelta1C causes filamentation, an effect that has been reported to result as well from the imbalance between FtsA+ and FtsZ+ molecules. These data indicate that the domain 1C of FtsA participates in the interaction of the protein with other FtsA molecules and with the other proteins that are incorporated at later stages of Ring assembly, and is not involved in the interaction with FtsZ and the localization of FtsA to the septal Ring. The deletion of the S12-S13 strands of domain 2B generates a protein (FtsADeltaS12-13) that retains the ability to interact with FtsA+. When the mutated protein is expressed at wild-type levels, it localizes into Division Rings and recruits FtsQ and FtsN, but it fails to sustain septation at normal levels resulting in filamentation. A fivefold overexpression of FtsADeltaS12-13 produces short cells that have normal Division Rings, but also cells with polar localization of the mutated protein, and cells with Rings at abnormal positions that result in the production of a fraction (15%) of small nucleoid-free cells. The S12-S13 strands of domain 2B are not essential for septation, but affect the localization of the Division Ring.

  • concentration and assembly of the Division Ring proteins ftsz ftsa and zipa duRing the escherichia coli cell cycle
    Journal of Bacteriology, 2003
    Co-Authors: Sonsoles Rueda, Miguel Vicente, Jesus Mingorance
    Abstract:

    The concentration of the cell Division proteins FtsZ, FtsA, and ZipA and their assembly into a Division Ring duRing the Escherichia coli B/r K cell cycle have been measured in synchronous cultures obtained by the membrane elution technique. Immunostaining of the three proteins revealed no organized structure in newly born cells. In a culture with a doubling time of 49 min, assembly of the Z Ring started around minute 25 and was detected first as a two-dot structure that became a sharp band before cell constriction. FtsA and ZipA localized into a Division Ring following the same pattern and time course as FtsZ. The concentration (amount relative to total mass) of the three proteins remained constant duRing one complete cell cycle, showing that assembly of a Division Ring is not driven by changes in the concentration of these proteins. Maintenance of the Z Ring duRing the process of septation is a dynamic energy-dependent event, as evidenced by its disappearance in cells treated with sodium azide.

Vitor O. Ferreira - One of the best experts on this subject based on the ideXlab platform.