This thesis examines localisations in the Weyl algebra. We investigate the injectives which characterise localisation via a number of dierent paths. The Weyl algebra does not exhibit the H-ring property which means that there exists an epimorphism between the injective envelopes of certain simple modules. We study these injective envelopes and other maps induced by the H-ring epimorphism. Localisations of the Weyl algebra are equivalent to universal localisations which are examples of homological ring epimorphisms. We are able to prove the torsion tilting classes associated with these tilting modules are generated by the kernel of the epimorphism between the injective envelopes of simple modules in the base and localised rings. These simple modules form simplification subcategories which are functorially finite and hence, have Auslander-Reiten sequences which are explicitly constructed.
Using the relationships between tilting modules and localisation we are able to prove that every torsion theory, (T ;F) in A1(k), is an Ore localisation,a question posed by Zhang. Duca asked when are two A1(k)-ideals, I a principal ideal and J an ideal with two generators, similar. We are able to provide a partial answer to this question using LeBruyn’s theory of canonical A1(k)-ideals.
Quivers are used to represent non-commutative algebras, and we describe the use and modify the quiver construction of Neeman, Ranicki and Schofield to demonstrate the usefulness of this technique.