The Isomorphism problem for incidence rings

Abstract

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 RR 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 {Pj | j Є (omega) } is any collection of (locally finite) preordered sets then there exists a ring S such that the incidence rings {I(Pj, S) | j Є (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).