(in Polish) Wymiary w teorii pierścieni 1000-1M18WTP
A few of the most important notions of dimension that play a prominent role in the theory of associative (but not necessarily commutative) rings will be presented. Namely, classical Krull dimension, Gelfand-Kirillov dimension, Gabriel-Rentschler dimension, and homological dimension.
The aim is to present the main properties of these dimensions, also in the context of fundamental notions and tools of the structural theory of rings (such as for example Goldie's theorem and the radical). We will discuss behavior of these dimensions under natural ring theoretic constructions. Some basic connections between these dimensions will be described. Examples of
applications and contexts in which these dimensions are usually used will be given. Some important open problems will be presented.
Type of course
Learning outcomes
Knows the definitions and the fundamental results describing the properties of the presented dimensions.
Knows the description of rings of small dimensions and can give examples illustrating possible values of dimensions.
Can formulate results describing the behavior of the dimensions under basic operations and algebraic constructions (such as: homomorphic images, subrings, polynomial rings, matrix rings).
Knows the relations between the presented dimensions in important classes of rings.
Can present examples of applications of various dimensions in ring theory.
Assessment criteria
Written final exam
Bibliography
S. Balcerzyk, T. Józefiak Commutative Rings.
G.R. Krause, T.H. Lenagan Growth of Algebras and Gelfand-Kirillov dimension.
J.C. McConnell, J.C.Robson Noncommutative Noetherian Rings.
J. Okniński Semigroup Algebras.
D.S. Passman A Course in Ring Theory.
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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