Associated prime ideals have long been known in commutative algebra for their center stage
role in the theory of primary decomposition. While the failure of uniqueness of primary de-
composition has been viewed by some as a significant drawback to this theory, the associated
primes that arise in this connection are independent of the decomposition. For this reason, as
well as the valuable resource they provide in homological methods, associated primes have come
to be viewed as objects worthy of study for their own sake, and not just as agents for primary
decomposition. These and other applications of associated primes in commutative ring theory
will be discussed in Chapter 1 .
More recently, associated primes have been introduced in noncommutative ring theory in
the hopes of extending the reach of this notion. In addition to primary decompositions over
noncommutative rings, associated primes facilitate the study of uniform modules over right
noetherian rings. Since uniform modules are in some sense ubiquitous, the importance of this
application cannot be overstated. Likewise, associated primes arise quite naturally in the study
of the structure of finitely generated modules over right noetherian rings.
The research philosophy guiding much of the work in this dissertation is the following: given
any result on associated primes in the commutative setting, can the result be extended to
noncommutative rings? Of particular interest to us is the question of how associated primes
behave under various types of ring extensions; in particular, polynomial extensions, matrix
extensions, and localizations. The case of polynomial extensions is especially noteworthy. C.