The generalized class group of a semiprime left noetherian ring
by
Christopher Pappacena
The Unversity of Colorado, Colorado Springs

In his thesis, G. Brookfield gives the construction of a group G_R that is defined for any (unital, associative) ring R. He also proves that, if R is semiprime left noetherian, then G_R embeds in a natural way as a subgroup of G₀(R). It turns out that the group operation for G_R is similar to that of Cl(R), the genus class group of R, when the latter is defined, and that the embedding of G_R in G₀(R) is similar to the embedding of Cl(R) into K₀(R).

In this paper, we explore the connection between G_R and Cl(R) when R is one of the rings studied in integral representation theory. We also show that, for certain classes of rings, G_R behaves very similarly to the class group. For example, under suitable hypotheses there exists ``Mayer-Vietoris" sequences for G<sub>R</sub>. These results show that, for these classes of rings, G<sub>R</sub> can be viewed as a kind of ``generalized class group."

Date received: November 9, 1998