Mathematical Sciences: Structure of Rings and Modules

本研究的目的是研究Noether环和仿射代数，特别是参考它们的表示理论。主要研究者将研究模的扩张及其在Notherian环的局部化中的应用。注意力将集中在Notherian环的链图上，以及该图中包含的信息如何预测模的可能扩张。有限环扩张的理想理论也将被考虑，希望利用这一领域的最新进展来获得关于环和有限扩张环的表示理论如何联系的结果。主要研究人员还将研究多项式有界增长的仿射代数，特别是Gelfand-Kirillov维度和相关的Poincare级数。在整个过程中，将强调构建和研究适当的范例。环是既有加法又有乘法的代数对象。这些对象自然而然地出现在数学的许多领域。给定一个环，一个重要的方面是包含原始环的环是否存在，它具有某些性质。何时可以局部化的问题就是这类问题，也是环论中的一个中心话题。

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.