Studies of rings with additional structure form one of the main branches of studies in the theory of associative rings and are conducted in many research centers in the world. There are many open problems in this area. Some of them are of substantial importance not only for ring theory or, more generally, for algebra but also for other branches of mathematics and other scientific disciplines, e.g., physics.
We intend to study the following classes of rings:
- Graded rings
- Rings with actions of some other structures, e.g. monoid and group actions, actions of Lie algebras or, more generally, actions of Hopf algebras.
- Rings which can be obtained from given rings via some constructions, e.g. polynomial rings or, more generally, semigroup rings, skew polynomial rings with endomorphisms and/or derivations (or, more generally, quantum groups), matrix rings, Cohn-Jordan extensions. etc.
There are many open problems in each of the above indicated areas. The aim of the studies in the project will be to examine some problems in the following three areas:
- Problems related to the structure of one-sided ideals and the prime spectrum of the above mentions classes of rings.
- Finiteness conditions and dimensions of rings and modules.
- Module theoretical problems concerning rings with additional structure.
Let us mention for instance Koethe’s problem, which was raised in 1930, which can be formulated as whether polynomial rings in one indeterminate over nil rings are jacobson radical or, equivalently, whether matrix rings over nil rings are nil. This problem inspires more general studies concerning for instance behaviour of radicals under the indicated, or similar, constructions. Many papers concerning that topic and also of the behaviour of different kind of ideals under such constructions were published in recent years. The intended studies of our project will in particular concern prime spectrum of rings and its behaviour under constructions.
We will also be concerned with the study of one-sided ideals. Let us mention for instance that the classical formulation of Koethe’s problem is whether two-sided ideals generated by one-sided nil ideals must be nil. Studies of one-sided ideal of rings very often employ tools of the theory of modules or lead to natural questions concerning modules. It is expected that some of our studies will deal with such questions.
Studying any ring theoretical problem might be easier when one assumes certain finiteness conditions, e.g. noetherianess. We intend to look at certain finiteness conditions, in particular dimensions (e.g., Goldie, Krull) of rings and modules (or even modular lattices) to study the given problems.
The above mentioned problems were studied by all participants involved into the project. It is exepected that an exchange of information, common discussions and research will benefit in finding some new methods or approaches, which should bring new original results. In particular the portuguese algebraists will get a new stimulation for extending their research areas and activity. As an outcome it is expected that a number of papers will be published in internationally recognized journals.