Spectrum and Ideal Theory of Chain Rings
by
Nikolai Dubrovin
Vladimir State University

It is known that the set of fractional principal ideals of a valuation ring of rank 1 (in the classical sense) constitute an Archimedean linearly ordered group; thus such a group can be enclosed in (R, +, 0, < ). How does look the monoid of ideals of chain domain with rank 1? That's the question which will be discussed in the talk. The talk is based on two works.

The first was made by the author in Canada in 1994 - "The rational closure of group rings of left-ordered groups" In particular, this work contains a method of embedding a group ring of the universal covering group of SL(2, R) in a valued skew field.

The second work "Classification of chain rings" was written together with Prof. H.H. Brungs and it is not published jet. In this paper all possible examples of chain domain with rank 1 are delivered. There exist three cases: - classical (as mentioned above), nearly simple (the only proper ideal coincides with Jacobson radical), exceptional (the case splits in a series of subcases). The third case is most interesting from the point of view of the theory of non-commutative rings. It can occur if and only if the prime ideal wich is not completely prime exists in the chain domain.

Date received: April 7, 1999