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 well‐known 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≤yf(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 one‐sided 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.