Weak incidence algebra and maximal ring of quotients

Abstract

Let X, X be two locally finite, preordered sets and let R be any indecomposable commutative ring. The incidence algebra I(X, R), in a sense, represents X, because of the wellknown result that if the rings I(X, R) and I(X ,R) are isomorphic, then X and X are isomorphic. In this paper, we consider a preordered set X that need not be locally finite but has the property that each of its equivalence classes of equivalent elements is finite. Define

Some chain conditions on weak incidence algebras

“…The algebra T is called weak incidence algebra of X over R. For a locally finite partially ordered set Y , the concept of incidence algebra I(Y ,R) is well known [6]. It can be proved on similar lines as for incidence algebras that for any two partially ordered sets X, Z and any two indecomposable commutative rings R, S, I * (X,R) and I * (Z,S) are isomorphic as rings if and only if X and Z are isomorphic and R and S are isomorphic [5]. It has been seen in [1,5] that weak incidence algebras can be used to construct rings whose left and right maximal rings of quotients need not be isomorphic.…”