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.

Bimodules over nest algebras and Deddens'

We generalise Dedden's Theorem for nest algebras to nest algebra bimodules. We define an object which extends the Jacobson radical of a nest algebra, and characterose it generalising a theorem of Erdos.

Simple components of the center of FG/J(FG)

Let G be a finite group, F a field, FG the group ring of G over F, and J(FG) the Jacobson radical of FG. Using a result of Berman and Witt, we give a method to determine the structure of the center of FG/J(FG), provided that F satisfies a field theoretical condition.

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.

On some Results Related to Köthe's Conjecture

The Köthe conjecture states that if a ring R has no nonzero nil ideals then R has no nonzero nil one-sided ideals. Although for more than 70 years significant progress has been made, it is still open in general. In this paper we survey some results related to the Köthe conjecture as well as some equivalent problems.

Highly efficient, stoichiometric radical exchange reactions using isoindoline profluorescent nitroxides

Exchange reactions between the isoindoline profluorescent nitroxide 1,1,3,3-tetramethyldibenzo[e,g]isoindolin-2-yloxyl (TMDBIO) and a TEMPO capped polystyrene were carried out. High conversions to the desired products were achieved using only stoichiometric ratios of nitroxide relative to polymer. The scope of this study was expanded by exploiting a di-nitroxide 9,10-bis(5-[1,1,3,3-tetramethylisoindolin-2-yloxy])anthracene (BTMIOA) as a connector between two polymer chains forming PS–nitroxide–PS systems.

Critical communications research in Australia : from radical populism to creative industries

This paper tracks the development of critical communicatiosn research in Australia over a 30 year period. It assesses the relative significance of critical theory, Marxist political economy and cultural studies to the development of such a tradition. it also evaluates the rise of 'creative industries' dicourse as an emergent development in the field, and a distinctive contribution of Australian media and communications research.

Prospective, non-randomized phase 2 clinical trial of carboplatin plus paclitaxel with sequential radical pelvic radiotherapy for uterine papillary serous carcinoma

Objective Uterine Papillary Serous Carcinoma (UPSC) is uncommon and accounts for less than 5% of all uterine cancers. Therefore the majority of evidence about the benefits of adjuvant treatment comes from retrospective case series. We conducted a prospective multi-centre non-randomized phase 2 clinical trial using four cycles of adjuvant paclitaxel plus carboplatin chemotherapy followed by pelvic radiotherapy, in order to evaluate the tolerability and safety of this approach. Methods This trial enrolled patients with newly diagnosed, previously untreated patients with stage 1b-4 (FIGO-1988) UPSC with a papillary serous component of at least 30%. Paclitaxel (175 mg/m2) and carboplatin (AUC 6) were administered on day 1 of each 3-week cycle for 4 cycles. Chemotherapy was followed by external beam radiotherapy to the whole pelvis (50.4 Gy over 5.5 weeks). Completion and toxicity of treatment (Common Toxicity Criteria, CTC) and quality of life measures were the primary outcome indicators. Results Twenty-nine of 31 patients completed treatment as planned. Dose reduction was needed in 9 patients (29%), treatment delay in 7 (23%), and treatment cessation in 2 patients (6.5%). Hematologic toxicity, grade 3 or 4 occurred in 19% (6/31) of patients. Patients' self-reported quality of life remained stable throughout treatment. Thirteen of the 29 patients with stages 1–3 disease (44.8%) recurred (average follow up 28.1 months, range 8–60 months). Conclusion This multimodal treatment is feasible, safe and tolerated reasonably well and would be suitable for use in multi-institutional prospective randomized clinical trials incorporating novel therapies in patients with UPSC.

Profluorescent nitroxides as sensitive probes of oxidative change and free radical reactions

This paper presents a review on the use of tethered nitroxide–fluorophore molecules as probes of oxidative change and free radical generation and reaction. The proximity of the nitroxide free radical to the fluorophore suppresses the normal fluorescence emission process. Nitroxide free radical scavenging, metabolism or redox chemistry return the system to its natural fluorescent state and so these tethered nitroxide–fluorophore molecules are described as being profluorescent. A survey of profluorescent nitroxides found in the literature is provided as well as background on the mechanism of action and applications of these compounds as fluorometric probes within the fields of biological, materials and environmental sciences.

Living characteristics of the free-radical ring-closing polymerization of diallyldimethylammonium chloride

In this communication we provide the most recent results on RAFT-mediated ring-closing polymerization of diallyldimethylammonium chloride (DADMAC). The polymerization was carried out in aqueous solution employing 2,2′-azobis(2-methylpropionamidine)-dihydrochloride as the free radical initiator and trithiocarbonate RAFT agent (2-{[(dodecylsulfanyl)carbonothioyl sulfanyl]}propanoic acid, DoPAT) as the controlling RAFT agent. The results show that – while the system is not as completely controlled as previously described – it is nevertheless possible to mediate the polymerization of DADMAC and impart some living characteristics onto the system. The initial study on the RAFT-mediated polymerization of DADMAC may have overestimated the degree of livingness within this reaction. However, it is possible – at low conversions – for some living characteristics to be observed, as the evolution of molecular weight with conversion is linear. In addition, polymers with a reasonably narrow polydispersity can be isolated.