Weak incidence algebra and maximal ring of quotients

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 I∗(X,R) to be the set of all those functions f : X ×X → R such that f(x,y) = 0, whenever x y and the set Sf of ordered pairs (x,y) with x < y and f(x,y) ≠ 0 is finite. For any f ,g ∈ I∗(X,R), r ∈ R, define f +g, fg, and rf in I∗(X,R) such that (f +g)(x,y) = f(x,y)+g(x,y), fg(x,y) = x z y f(x,z)g(z,y), rf(x,y) = r · f(x,y). This makes I∗(X,R) an R-algebra, called the weak incidence algebra of X over R. In the first part of the paper it is shown that indeed I∗(X,R) represents X. After this all the essential onesided ideals of I∗(X,R) are determined and the maximal right (left) ring of quotients of I∗(X,R) is discussed. It is shown that the results proved can give a large class of rings whose maximal right ring of quotients need not be isomorphic to its maximal left ring of quotients.