The following are "alternative abstracts" for my papers and preprints. They are aimed to be readable to mathematicians who have a passing familiarity with the language of abstract algebra, but who are not algebraists. Please let me know whether I've succeeded!
Several celebrated papers in the '70s, perhaps beginning with the landmark paper of Roseblade, answered long-standing questions about the prime ideal structure of polycyclic group algebras R = k[G]. One of the most fundamental tools of Roseblade's analysis was a clear description of the elements in certain quotient rings, R/I, which were centralised by some large subgroup of G. The proof of this result is very elementary in this classical "discrete" scenario, but the proof does not port cleanly to the "continuous" world of completed group algebras. Here we show that an earlier argument by Ardakov can be adapted to prove an appropriate analogue in the modern setting.
Iterated local skew power series rings are rings that arise very naturally in algebra, representation theory and number theory: intuitively, they correspond to completions of (super)soluble objects (like group algebras or universal enveloping algebras) or of noncommutative polynomial constructions (like skew polynomial rings). We discuss several natural examples, and show that many of them in fact obey a further triangularity condition, mimicking the role of nilpotent groups or supersoluble Lie algebras within the soluble setting. We calculate the dimension theory of these rings, and show that they do indeed behave as this intution would have us expect.
Catenarity is a well-behavedness criterion on lengths of chains of prime ideals in a ring. Due to celebrated results of the latter half of the 20th century, many classical "discrete" algebraic objects are now known to be catenary, including certain group algebras and universal enveloping algebras. We bring the theory up to date in a more modern "complete" setting.
One incredibly common technique to understand the prime ideals of a "nasty" ring S is: find a "nice" subring R, and try to pass information between R and S. For instance: (1) If B is a prime ideal of S, what properties does its contraction to (= intersection with) R satisfy? (2) If A is an ideal of R, when is its extension to S a prime ideal? Question (1) is usually easier, but in this context, it is only a stepping stone to the real question of interest, question (2). We work in a context of certain "nasty" rings S of interest, choose an explicitly defined class of "nice" subrings R, and solve question (2) in this context.
Prime rings are usually far nicer to deal with than non-prime rings. There are standard ways of producing a prime ring from a non-prime ring: simply pass to a prime homomorphic image ("factor") of the non-prime ring. But this raises several questions:
(1) How much information do we lose about the original ring by doing so? (The larger the image, the less information we lose, so we focus on maximal images here.)
(2) What does the resulting ring look like? (Does it have a nice, down-to-earth description that we can actually use to calculate things in the ring?)
(3) How much information can we infer about the original ring from the resulting ring? We give precise answers to these questions for a large class of interesting rings, namely all Iwasawa algebras over a finite base field.
The key to understanding complicated objects is usually to break them up into pieces that are better-understood. We work with a large class of infinite groups. By choosing our pieces carefully, we show that a lot of problems can be solved for the groups in this class by turning them into much simpler problems in linear algebra, and in fact that this linear algebra is usually just multiplying by scalars.