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Dr. Scott Annin Research Statement
 

Summary: Dr. Scott Annin
Research Statement
January 2011
I have been involved in a number of different research projects during my undergraduate and
graduate studies, and now as a faculty member at Cal State Fullerton (CSUF). To various
extents, these studies have broadly intersected with ring theory, group theory, semigroup
theory, linear algebra, combinatorics, and mathematics education.
My Ph.D. dissertation [3] was in the area of noncommutative ring theory. One of my
specific interests in that area is the associated prime ideals of a module over a ring. These
ideals have long played an important role in commutative rings, where they enjoy a rich
and extensive theory and applications, most notably to the well-known theory of primary
decomposition in computational algebra. At the same time, there is a useful and interesting
dual theory concerning "attached" prime ideals. These ideals were first introduced in 1973
by I.G. Macdonald [15] in the context of representable modules. The theory was somewhat
restrictive, applying only to a special class of modules over a commutative ring. In my
dissertation, I generalized the theory to arbitrary modules over (possibly) noncommutative
rings. In fact, my dissertation contributed heavily to the generalization of the theory of
both associated and attached primes and their properties to noncommutative ring theory.
The issue of ascertaining how various ring-theoretic and module-theoretic concepts (like
associated and attached prime ideals) behave under various types of change of rings, such as

  

Source: Annin, Scott - Department of Mathematics, California State University, Fullerton

 

Collections: Mathematics