Mathematical Sciences: Structure of Rings and Modules

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The aim of this research is to study Noetherian rings and affine algebras, with special reference to their representation theory. The principal investigator will study extensions of modules with applications to localizations of Noetherian rings. Attention will be focused on the graph of links of a Noetherian ring, and how information contained in this graph can predict the possible extensions of a module. The ideal theory of finite ring extensions will also be considered in the hopes of using recent progress in this area to obtain results on how the representation theories of a ring and a finite extension ring are related. The principal investigator will also study affine algebras of polynomial bounded growth, with special reference to the Gelfand- Kirillov dimension and the associated Poincare series. Throughout there will be an emphasis on the construction and study of appropriate examples. A ring is an algebraic object having both an addition and a multiplication. These objects arise naturally in many areas of mathematics. Given a ring, one area of importance is if a ring containing the original one exists having certain properties. The question of when it is possible to localize is of this type and is a central topic in ring theory.