Non-Commutative Ring Theory
Hisaya Tsutsui (NIU):
Non-commutative ring theory
Specific interests include prime and primitive Noetherian rings and injective modules.
My research interests are in the area of non-commutative ring theory. A ring is a set on which two binary operations are defined. Concrete examples of such an abstract model include the set of integers with usual addition and multiplications, and the set of n by n matrices over a ring with the well-known addition and the multiplication. The latter example is an example of a non-commutative ring: the multiplication of two matrices does not commute.
A ring has DNA like objects called ideals. I have investigated the structure of rings whose ideals have particular properties. There are some classes of rings that I am particularly interested: the class of von Neumann regular rings, right Noetherian rings, and their generalizations. Important examples of these rings include: rings of linear transformations, certain rings of differential operators, and universal enveloping algebras of finite dimensional Lie algebras, which plays an essential role in quantum field theory and atomic particle physics.
Normally, an entry level introduction to ring theory is first given to undergraduate mathematics majors after their completion of the calculus and linear algebra sequence.