Surjeet Singh and Fawzi Al-Thukair

Copyright © 2004 Surjeet Singh and Fawzi Al-Thukair. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.