The Isomorphism Problem for Incidence Rings
by
Gene Abrams
University of Colorado at Colorado Springs
Coauthors: Jeremy Haefner, Angel del Rio

Let P and P' be finite preordered sets, and let R be a ring for which the number of nonzero summands in a direct decomposition of the regular module R_R is bounded. We show that if the incidence rings I(P, R) and I(P', R) are isomorphic as rings, then P and P' are isomorphic as preordered sets. We give a stronger version of this result in case P and P' are partially ordered. We show that various natural extensions of these results fail. Specifically, we show that if {P_j | j in \Omega} is any collection of (locally finite) preordered sets then there exists a ring S such that the incidence rings {I(P_j, S) | j in \Omega} are pairwise isomorphic. Additionally, we verify that there exists a finite dimensional algebra R and locally finite, nonisomorphic partially ordered sets P and P' for which I(P, R) =~ I(P', R).

Date received: October 23, 1998