Prime Segments in Simple Artinian Rings
by
Enver Osmanagic
University of Alberta
Coauthors: H.H.Brungs (University of Alberta,Canada), H.Marubayashi (Naruto University of Education, Japan)

This talk is based on the joint work with H.H. Brungs and H. Marubayashi.

A Dubrovin valuation ring in a simple artinian ring A is a Bezout order R of A so that R/J(R) is a simple artinian ring. Prime ideals P of R so that R/P is a prime Goldie ring are in 1-1 correspondence with overrings of R in A. Such ideals are called Goldie primes and a prime segment of A is a pair P₁ contains P₂ of neighbouring Goldie primes of R.

There are exactly three types of prime segments:

a) The prime segment P₁ contains P₂ is archimedean, i.e., for all a in P₁\P₂ there exists an ideal I subset or equal P₁ such that the intersection of all powers of I is equal P₂.

b) The prime segment P₁ contains P₂ is simple, i.e., no ideals between P₁ and P₂.

c) The prime segment P₁ contains P₂ is exceptional, i.e., there exists a prime ideal Q that is not Goldie prime, P₁ contains Q contains P₂, and no ideals between P₁ and Q.

The type of the prime segment P₁ contains P₂ is the same for any Dubrovin valuation ring R of A that contains this prime segment.

In the special case of the ring R of rank one (J(R) and (0) are the only Goldie primes), the structure of the ideal lattice of R is completely described, using the fact that such ring is a maximal order in A and that divisorial R-ideals of A form a group.

Date received: April 30, 1999