FREE GROUP ALGEBRAS IN DIVISION RINGS

Let k be an algebraically closed field of characteristic zero and let L be an algebraic function field over k. Let sigma : L -> L be a k-automorphism of infinite order, and let D be the skew field of fractions of the skew polynomial ring L[t; sigma]. We show that D contains the group algebra kF of the free group F of rank 2.

A SURVEY ON FREE OBJECTS IN DIVISION RINGS AND IN DIVISION RINGS WITH AN INVOLUTION

Let D be a division ring with center k, and let D-dagger be its multiplicative group. We investigate the existence of free groups in D-dagger, and free algebras and free group algebras in D. We also go through the case when D has an involution * and consider the existence of free symmetric and unitary pairs in D-dagger.

A NOTE ON FREE GROUPS IN THE RING OF FRACTIONS OF SKEW POLYNOMIAL RINGS

Let L be a function field over the rationals and let D denote the skew field of fractions of L[t; sigma], the skew polynomial ring in t, over L, with automorphism sigma. We prove that the multiplicative group D(x) of D contains a free noncyclic subgroup.

Class Notes for Math 901/902: Abstract Algebra, Instructor Tom Marley

Topics include: Free groups and presentations; Automorphism groups; Semidirect products; Classification of groups of small order; Normal series: composition, derived, and solvable series; Algebraic field extensions, splitting fields, algebraic closures; Separable algebraic extensions, the Primitive Element Theorem; Inseparability, purely inseparable extensions; Finite fields; Cyclotomic field extensions; Galois theory; Norm and trace maps of an algebraic field extension; Solvability by radicals, Galois' theorem; Transcendence degree; Rings and modules: Examples and basic properties; Exact sequences, split short exact sequences; Free modules, projective modules; Localization of (commutative) rings and modules; The prime spectrum of a ring; Nakayama's lemma; Basic category theory; The Hom functors; Tensor products, adjointness; Left/right Noetherian and Artinian modules; Composition series, the Jordan-Holder Theorem; Semisimple rings; The Artin-Wedderburn Theorem; The Density Theorem; The Jacobson radical; Artinian rings; von Neumann regular rings; Wedderburn's theorem on finite division rings; Group representations, character theory; Integral ring extensions; Burnside's paqb Theorem; Injective modules.

Structure theorems for noncommutative complete local rings

If R is a ring with identity, let N(R) denote the Jacobson radical of R. R is local if R/N(R) is an artinian simple ring and ∩N(R)ⁱ = 0. It is known that if R is complete in the N(R)-adic topology then R is equal to (B)_n, the full n by n matrix ring over B where E/N(E) is a division ring. The main results of the thesis deal with the structure of such rings B. In fact we have the following.

If B is a complete local algebra over F where B/N(B) is a finite dimensional normal extension of F and N(B) is finitely generated as a left ideal by k elements, then there exist automorphisms g_i,...,g_k of B/N(B) over F such that B is a homomorphic image of B/N[[x₁,…,x_k;g₁,…,g_k]] the power series ring over B/N(B) in noncommuting indeterminates x_i, where x_ib = g_i(b)x_i for all b ϵ B/N.

Another theorem generalizes this result to complete local rings which have suitable commutative subrings. As a corollary of this we have the following. Let B be a complete local ring with B/N(B) a finite field. If N(B) is finitely generated as a left ideal by k elements then there exist automorphisms g₁,…,g_k of a v-ring V such that B is a homomorphic image of V [[x₁,…,x_k;g₁,…,g_k]].

In both these results it is essential to know the structure of N(B) as a two sided module over a suitable subring of B.

Spatiotemporal mechanisms for actomyosin ring assembly and contraction in budding yeast cell division

Dissertation presented to obtain the Ph.D degree in Molecular Medicine

Aggregates in the strength and gravity regime: Particles sizes in Saturn's rings

Particles in Saturn's main rings range in size from dust to kilometer-sized objects. Their size distribution is thought to be a result of competing accretion and fragmentation processes. While growth is naturally limited in tidal environments, frequent collisions among these objects may contribute to both accretion and fragmentation. As ring particles are primarily made of water ice attractive surface forces like adhesion could significantly influence these processes, finally determining the resulting size distribution. Here, we derive analytic expressions for the specific self-energy Q and related specific break-up energy Q(star) of aggregates. These expressions can be used for any aggregate type composed of monomeric constituents. We compare these expressions to numerical experiments where we create aggregates of various types including: regular packings like the face-centered cubic (fcc), Ballistic Particle Cluster Aggregates (BPCA), and modified BPCAs including e.g. different constituent size distributions. We show that accounting for attractive surface forces such as adhesion a simple approach is able to: (a) generally account for the size dependence of the specific break-up energy for fragmentation to occur reported in the literature, namely the division into "strength" and "gravity" regimes and (b) estimate the maximum aggregate size in a collisional ensemble to be on the order of a few tens of meters, consistent with the maximum particle size observed in Saturn's rings of about 10 m. (c) 2012 Elsevier B.V. All rights reserved.

RefZ Facilitates the Switch from Medial to Polar Division during Spore Formation in Bacillus subtilis

During sporulation, Bacillus subtilis redeploys the division protein FtsZ from midcell to the cell poles, ultimately generating an asymmetric septum. Here, we describe a sporulation-induced protein, RefZ, that facilitates the switch from a medial to a polar FtsZ ring placement. The artificial expression of RefZ during vegetative growth converts FtsZ rings into FtsZ spirals, arcs, and foci, leading to filamentation and lysis. Mutations in FtsZ specifically suppress RefZ-dependent division inhibition, suggesting that RefZ may target FtsZ. During sporulation, cells lacking RefZ are delayed in polar FtsZ ring formation, spending more time in the medial and transition stages of FtsZ ring assembly. A RefZ-green fluorescent protein (GFP) fusion localizes in weak polar foci at the onset of sporulation and as a brighter midcell focus at the time of polar division. RefZ has a TetR DNA binding motif, and point mutations in the putative recognition helix disrupt focus formation and abrogate cell division inhibition. Finally, chromatin immunoprecipitation assays identified sites of RefZ enrichment in the origin region and near the terminus. Collectively, these data support a model in which RefZ helps promote the switch from medial to polar division and is guided by the organization of the chromosome. Models in which RefZ acts as an activator of FtsZ ring assembly near the cell poles or as an inhibitor of the transient medial ring at midcell are discussed.

Interaction between cell division proteins FtsE and FtsZ.

FtsE and FtsX, which are widely conserved homologs of ABC transporters and interact with each other, have important but unknown functions in bacterial cell division. Coimmunoprecipitation of Escherichia coli cell extracts revealed that a functional FLAG-tagged version of FtsE, the putative ATP-binding component, interacts with FtsZ, the bacterial tubulin homolog required to assemble the cytokinetic Z ring and recruit the components of the divisome. This interaction is independent of FtsX, the predicted membrane component of the ABC transporter, which has been shown previously to interact with FtsE. The interaction also occurred independently of FtsA or ZipA, two other E. coli cell division proteins that interact with FtsZ. In addition, FtsZ copurified with FLAG-FtsE. Surprisingly, the conserved C-terminal tail of FtsZ, which interacts with other cell division proteins, such as FtsA and ZipA, was dispensable for interaction with FtsE. In support of a direct interaction with FtsZ, targeting of a green fluorescent protein (GFP)-FtsE fusion to Z rings required FtsZ, but not FtsA. Although GFP-FtsE failed to target Z rings in the absence of ZipA, its localization was restored in the presence of the ftsA* bypass suppressor, indicating that the requirement for ZipA is indirect. Coexpression of FLAG-FtsE and FtsX under certain conditions resulted in efficient formation of minicells, also consistent with an FtsE-FtsZ interaction and with the idea that FtsE and FtsX regulate the activity of the divisome.

Temporal and spatial regulation of cell division by the nucleoid in Escherichia coli

Selection of division sites and coordination of cytokinesis with other cell cycle events are critical for every organism to proliferate. In E. coli, the nucleoid is proposed to exclude division from the site of the chromosome (nucleoid occlusion model). We studied the effect of the nucleoid on timing and placement of cell division. An early cell division protein, FtsZ, was used to follow development of the division septum. FtsZ forms a ring structure (Z ring) at potential division sites. The dynamics of Z ring was visualized in live cells by fusing FtsZ with a green fluorescent protein (GFP). Emanating FtsZ-GFP polymers from the constricted septum or aggregates in daughter cells were also observed, probably representing the FtsZ depolymerization and immature FtsZ nucleation processes. We next examined the nucleoid occlusion model. Mutants carrying abnormally positioned chromosomes were employed. In chromosomal partition mutants, replicated chromosomes cannot segregate. The Z ring was excluded from midcell to the edge of the nucleoid. This negative effect of nucleoids was further confirmed in replication deficient dnaA mutants, in which only a single chromosome is present in the cell center. These results suggest that the nucleoid, replicating or not, inhibits division in the area where the chromosome occupies. In addition, increasing the level of FtsZ does not overcome nucleoid inhibition. Interestingly in anucleate cells produced by both mutants, the Z ring was localized in the central part of the cell, which indicates that the nucleoid is not required for FtsZ assembly. Relaxation of chromosomes by reducing the gyrase activity or disruption of protein translation/translocation did not abolish the division inhibition capacity of the nucleoid. However, preventing transcription did compromise the nucleoid occlusion effect, leading to formation of multiple FtsZ rings above the nucleoid. In summary, we demonstrate that nucleoids negatively regulate the timing and position of division by inhibiting FtsZ assembly at unselected sites. Relief of this inhibition at midcell is coincident with the completion of DNA replication. On the other hand, FtsZ assembly does not require the nucleoid. ^

Identifying the roles of the Escherichia coli FtsZ-associated proteins by minimizing the requirements for division

Formation of the FtsZ ring (Z ring) in Escherichia coli is the first step in assembly of the divisome, a molecular machine composed of 14 known proteins which are all required for cell division. Although the biochemical functions of most divisome proteins are unknown, several of these have overlapping roles in ensuring that the Z ring assembles at the cytoplasmic membrane and is active. ^ We identified a single amino acid change in FtsA, R286W, renamed FtsA*, that completely bypasses the requirement for ZipA in cell division. This and other data suggest that FtsA* is a hyperactive form of FtsA that can replace the multiple functions normally assumed by ZipA, which include stabilization of Z rings, recruitment of downstream cell division proteins, and anchoring the Z ring to the membrane. This is the first example of complete functional replacement of an essential prokaryotic cell division protein by another. ^ Cells expressing ftsA* with a complete deletion of ftsK are viable and divide, although many of these ftsK null cells formed multiseptate chains, suggesting a role in cell separation for FtsK. In addition, strains expressing extra ftsAZ, ftsQ, ftsB, zipA or ftsN, were also able to survive and divide in the absence of ftsK. The cytoplasmic and transmembrane domains of FtsQ were sufficient to allow viability and septum formation to ftsK deleted strains. These findings suggest that FtsK is normally involved in stabilizing the divisome and shares functional overlap with other cell division proteins. ^ As well as permitting the removal of other divisome components, the presence of FtsA* in otherwise wild-type cells accelerated Z-ring assembly, which resulted in a significant decrease in the average length of cells. In support of its role in Z-ring stability, FtsA* suppressed the cell division inhibition caused by overexpressing FtsZ. FtsA* did not affect FtsZ turnover within the Z ring as measured by fluorescence recovery after photobleaching. Turnover of FtsA* in the ring was somewhat faster than wild-type FtsA. Yeast two-hybrid data suggest that FtsA* has an increased affinity for FtsZ relative to wild-type FtsA. These results indicate that FtsA* interacts with FtsZ more strongly, and its enhancement of Z ring assembly may explain why FtsA* can permit survival of cells lacking ZipA or FtsK.^

Low cost time division multiplexing of identical optical fibre ring intensity sensors

A time division multiplexing (TDM) array for passive multiplexing of identical fibre, optic intensity sensors has been demonstrated. Microbending loss sensors are introduced in fibre optic rings and pressure information is directly detected, demultiplexed and demodulated from the relative amplitude of the first two pulses produced on each ring. Several dynamic ranges from 6 dB to 14 dB are shown. A comparison between both fibre optic ring and Mach-Zehnder structure impulse responses is carried out and the consequences derived from second- and higher-order recirculating ring pulses are also evaluated. This technique can be applied to those TDM intensity sensing schemes which require low cost, high number of identical sensors, and suffer high element loss and undersirable intensity fluctuations at low frequencies.

The MinC component of the division site selection system in Escherichia coli interacts with FtsZ to prevent polymerization

Positioning of the Z ring at the midcell site in Escherichia coli is assured by the min system, which masks polar sites through topological regulation of MinC, an inhibitor of division. To study how MinC inhibits division, we have generated a MalE-MinC fusion that retains full biological activity. We find that MalE-MinC interacts with FtsZ and prevents polymerization without inhibiting FtsZ's GTPase activity. MalE-MinC19 has reduced ability to inhibit division, reduced affinity for FtsZ, and reduced ability to inhibit FtsZ polymerization. These results, along with MinC localization, suggest that MinC rapidly oscillates between the poles of the cell to destabilize FtsZ filaments that have formed before they mature into polar Z rings.

Bacterial cell division protein FtsZ assembles into protofilament sheets and minirings, structural homologs of tubulin polymers.

The bacterial cell division protein FtsZ is a homolog of tubulin, but it has not been determined whether FtsZ polymers are structurally related to the microtubule lattice. In the present study, we have obtained high-resolution electron micrographs of two FtsZ polymers that show remarkable similarity to tubulin polymers. The first is a two-dimensional sheet of protofilaments with a lattice very similar to that of the microtubule wall. The second is a miniring, consisting of a single protofilament in a sharply curved, planar conformation. FtsZ minirings are very similar to tubulin rings that are formed upon disassembly of microtubules but are about half the diameter. This suggests that the curved conformation occurs at every FtsZ subunit, but in tubulin rings the conformation occurs at either beta- or alpha-tubulin subunits but not both. We conclude that the functional polymer of FtsZ in bacterial cell division is a long thin sheet of protofilaments. There is sufficient FtsZ in Escherichia coli to form a protofilament that encircles the cell 20 times. The similarity of polymers formed by FtsZ and tubulin implies that the protofilament sheet is an ancient cytoskeletal system, originally functioning in bacterial cell division and later modified to make microtubules.