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 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 Ω} 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 Ω} 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).

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