International Journal of Mathematics and Mathematical SciencesVolume 2004 (2004), Issue 53, Pages 2835-2845doi:10.1155/S0161171204311130

# Surjeet Singh and Fawzi Al-Thukair

Department of Mathematics, King Saud University, PO Box 2455, Riyadh 11451, Saudi Arabia

Received 12 November 2003

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×XR 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,gI*(X,R), rR, 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)=xzyf(x,z)g(z,y), rf(x,y)=rf(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.