Mathematical Sciences: Non-Commutative Ring Theory

本项目涉及非交换环理论的各个领域。首席研究员将研究PI环及其表示，重点是线性群的应用，规范化元素的行为以及PI环在矩阵中的可嵌入性。具有Gelfand-Kirillov维数为2的仿射素环也将被研究，目的是确定它们何时是原始的。最后，讨论了非交换代数上约化群的不变量环。本研究属于环理论的一般范畴。环是一个代数对象，在它上面定义了加法和乘法。虽然加法运算满足交换律，但乘法运算不需要满足交换律。一个乘法不可交换的环的例子是整数上的nxn个矩阵的集合。非交换环的研究由于其对其他数学和物理分支的重要意义而成为代数的重要组成部分。

This project is concerned with various areas of noncommutative ring theory. The principal investigator will study PI rings and their representations focusing on applications to linear groups, the behavior of normalizing elements and the embeddability of PI rings into matrices. Affine prime rings with Gelfand-Kirillov dimension two will also be investigated with the goal of determining when they are primitive. Finally, rings of invariants of reductive groups on certain noncommutative algebras will be considered. This research is in the general area of ring theory. A ring is an algebraic object having both an addition and a multiplication defined on it. Although the additive operation satisfies the commutative law, the multiplicative operation is not required to do so. An example of a ring for which multiplication is not commutative is the collection of nxn matrices over the integers. The study of noncommutative rings has become an important part of algebra because of its increasing significance to other branches of mathematics and physics.