Left saturation closure : Theory and algorithms
Hoffmann, Johannes; Levandovskyy, Viktor (Thesis advisor); Zerz, Eva Barbara (Thesis advisor)
Dissertation / PhD Thesis
Ore localization of rings and modules is a technique that is widely used throughout non-commutative algebra with applications in ring theory, dimension theory, non-commutative geometry, and many other areas. In this thesis, we pursue several goals. First of all, we give a thorough introduction to Ore localization, starting from its axiomatic definition and working our way through the classical construction inspired by Ore's original work. While Ore localization is perfectly compatible with the addition operation of the ring, the presence of an additive structure is in no way required for the construction. With this in mind, we explore how the theory of Ore localization changes when we consider monoids instead of rings. The main contribution of this thesis is the notion of left saturation closure or LSat and the applications thereof: for a left denominator set S in an arbitrary ring R, LSat(S) represents a canonical form of S with respect to R-fixing isomorphisms of localizations of R. Furthermore, LSat(S) is involved in many characterizations of left invertible elements, left ideals, and other structural properties of the localization S^(-1)R. We show that LSat(S) is a left denominator set in R if and only if S^(-1)R is Dedekind-finite, in which case S^(-1)R is canonically isomorphic to LSat(S)^(-1)R and LSat(S) characterizes the unit group of S^(-1)R. We study classes of rings that satisfy this condition: for Ore domains we characterize maximal and pre-maximal left denominator set, and for factorization domains we prove that any saturated left denominator set is uniquely determined by the set of irreducible elements it contains. As an illustration we discuss several left denominator sets in Weyl algebras and their corresponding localizations. Afterwards, we expand our structural investigations to Ore-localized modules, where we recognize the well-known concept of local closure of a submodule as another special case of left saturation closure. We identify local torsion, which is a generalization of the classical notion of torsion, as a special case of local closure and show further connections between the two concepts. In the last part of this thesis, we give algorithmic solutions to several problems related to Ore localizations of G-algebras, or OLGAs for short. We describe our framework to perform basic arithmetic in OLGAs as well as our implementation in the computer algebra system Singular:Plural. Lastly, we develop multiple algorithms to compute local closure in important special cases.