Dimensions in Ring Theory 1000-1M24WTP
The lecture will be devoted to the presentation of several the most important concepts of dimension, which play a particularly important role, as a tool, in the theory of associative (generally non-commutative) rings. Namely:
• classic Krull dimension,
• Gelfand-Kirillov dimension,
• Goldie dimension,
• Gabriel-Rentschler dimension,
• dimensions of a homological nature (projective, injective, global, etc.).
The aim is to present basic properties of these dimensions, also in the context of fundamental concepts and tools of structural ring theory (such as, for instance, Goldie's theorem or various types of radicals). We will discuss behaviour of these dimensions in relation to certain important constructions of ring theory. We will also provide explanations for some relations holding between these dimensions, examples of their applications, and contexts in which they are used. Finally, we will present some important open problems regarding the discussed topics.
Type of course
Prerequisites
Prerequisites (description)
Course coordinators
Learning outcomes
Student:
• knows the definitions of: Krull, Gelfand-Kirillov, Goldi, Gabriel-Rentschler and homological dimensions.
• is able to use the concepts used in these definitions, namely: growth function, prime ideals, the concept of uniform module, free module, projective, injective module and the related resolvents.
• is able to justify the correctness of the definition of the Gelfand-Kirillov and homological dimensions.
• is able to characterise classes of rings of low dimensions (of dimension 0 and/or of dimension 1).
• knows the set of possible values for each dimension and is able to give examples of rings for which a given dimension has a given value.
• can provide examples of classes of rings for which certain dimensions coincide, as well as examples of rings for which different dimensions take different values.
• knows the behaviour of dimensions in relation to basic operations and algebraic constructions: passing to a subring, to a homomorphic image, to a matrix ring, to a ring of polynomials, to a finite extension (in the sense of modules); is able to reason to justify appropriate statements.
• knows and is able to apply theorems allowing to calculate dimensions of finitely generated commutative algebras (Noether's normalisation theorem, theorems on the behaviour of prime ideals under integer extensions).
• is able to indicate contexts in which particular dimensions are used and provide examples of such applications.
Assessment criteria
attendance and activity in class; oral exam
Bibliography
1. S. Balcerzyk, T. Józefiak, Pierścienie Przemienne.
2. G. R. Krause, T. H. Lenagan, Growth of Algebras and Gelfand-Kirillov dimension.
3. T. Y. Lam, Lectures on Modules and Rings.
4. J. C. McConnell, J. C. Robson, Noncommutative Noetherian Rings.
5. C. Nǎstǎsescu, F. van Oystaeyen, Dimensions of Ring Theory.
6. J. Okniński, Semigroup Algebras.
7. D. S. Passman, A Course in Ring Theory.
8. C. Weibel, An Introduction to Homological Algebra.
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:
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