Title:

On asymptotic stability of prime ideals in noncommutative rings

This thesis is concerned with the asymptotic behaviour of the prime divisors of powers of an ideal I in a ring R. The main focus is on the case when R is a prime Noetherian polynomial identity ring. Our starting point is a theorem, first proved by M. Brodrnann, which shows that in a commutative Noetherian ring, the prime divisors of powers of an ideal turn out, to be asymptotically stable. We first, generalize this result to Azumaya algebras, and then attempt to find an overring which satisfies the following two criteria: 1. prime divisors 'go up and down' between this overring and the original ring; 2. prime divisors 'go up and down' between this overring and its centre. After finding conditions under which such an overring exists, it is possible to formulate a theorem concerning asymptotic stability of prime divisors in the original ring, using the fact that prime divisors are asymptotically stable in the centre of the overring. In chapter 1 we outline the basic definitions, results and theory required for this thesis. In chapter 2 we first establish some properties concerning associated primes of a module, and in Proposition 2.1.3 we note that a result connecting two sequences, due to M. Brodinann in the commutative case, is equally valid in the noncomrnutative setting. Next we outline a proof of the definitive asymptotic stability result in the commutative theory, due to M. Brodinann. The proof followed in this thesis is that given by S. McAdam and P. Eakin. In the final section of this chapter we discuss some results concerning invertible ideals, due to A.J. Gray, and their applications to aspects of the proof of M. Brodmann's result. We note difficulties with this approach and give reasons why this does not lead to a generalization of M. Brodmann's theorem. In chapter 3 we provide the generalization of M. Brodmann's theorem to Azumaya algebras. Affiliated prime ideals have often been viewed as a generalized class of associated prime ideals in the noncommutative setting, which contain more information than associated prime ideals. With this in mind, we shift the focus of our study from associated prime ideals to affiliated prime ideals. We also show that there is a very simple relationship between the prime divisors of an ideal in an Azuinaya algebra Rand the prime divisors of the corresponding ideal of the centre of It. We give an example to illustrate this result. In chapter 4 we introduce localizations of rings and modules, giving basic properties. We give a method of 'going up and down' between ideals of the original ring and ideals of localizations of that ring, and give conditions under which this method is applicable. We then use this to derive a relationship between the prime divisors of the original ring and the prime divisors of localizations of that ring. In chapter 5 we attempt to find a ring satisfying the two criteria set out above, using a localization as the overring. We give conditions under which this localization is an Azumaya algebra, enabling us to make use of the results in chapter 3. 1* iist, we give necessary and sufficient conditions for asymptotic stability of prime divisors to hold in a prime Noetherian polynomial identity ring. We then give alternative sufficient conditions for asymptotic stability to hold in the same class of lings. We conclude with a discussion of some results in the general Noetherian case.
