Still under construction.
The members of the algebra group who work in this area are
Allen Bell and
Ian Musson.
This description is intended for those have already had some exposure to ring theory; for a brief description of rings, read this.
The main thrust of the theory of commutative rings is intimately related to the theory of rings of polynomial functions (and rings derived from them such as quotients and localizations). Such rings are noetherian, that is, every ascending chain of ideals eventually becomes stationary. [For non-commutative rings, we must assume this not just for two-sided ideals but for one-sided ideals as well.]
Non-commutative rings are a much more varied species, but they tend to be related to rings of linear operators, which unlike functions, do not commute with each other. The study of non-commutative rings is a field begun in the 20th century, and much of the early work concentrated on division rings and algebras that were finite dimensional over a field. Such algebras are always artinian, that is, every descending chain of left (or right) ideals eventually becomes stationary. An artinian ring is always noetherian, but the converse is not true: artinian rings form a much more restricted class. The concepts of noetherian and artinian rings were abstracted from specific commutative examples in the 1920's. Natural examples of non-commutative rings need not be noetherian; nevertheless, the noetherian hypothesis is very useful and fortunately does hold in many cases.
It is well-known that any commutative integral domain has a field of fractions; this need not be true for a non-commutative ring lacking zero divisors. O. Ore in 1931 gave necessary and sufficient conditions for a division ring of fractions to exist. While many interesting ring theoretic results were proven in between, it is probably fair to say that the modern study of non-commutative noetherian rings began with A. W. Goldie's work in 1958-1960 giving necessary and sufficient conditions for a ring to have a semisimple ring of fractions. This result is much deeper than Ore's 1931 result, and the techniques and subsidiary results gave researchers technical tools that are still in use today. [Goldie showed a noetherian ring has a semisimple ring of fractions if and only if it is semiprime and it has a division ring of fractions if and only if it has no zero divisors.]
Here are some examples of non-commutative noetherian rings that have been studied by members of the algebra group:
Here are some topics connected with noetherian rings that have been studied by members of the algebra group: