Department of Mathematics, King Saud University, PO Box 2455, Riyadh 11451, Saudi Arabia
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.
Let , be two locally finite, preordered sets and let be any indecomposable commutative ring. The incidence algebra , in a sense, represents , because of the well-known result that if the rings and are isomorphic, then and are isomorphic. In this paper, we consider a preordered set that need not be locally finite but has the property that each of its equivalence classes of equivalent elements is finite. Define to be the set of all those functions such that , whenever and the set of ordered pairs with and is finite. For any , , define , , and in such that , , . This makes an -algebra, called the weak incidence algebra of over . In the first part of the paper it is shown that indeed represents . After this all the essential one-sided ideals of are determined and the maximal right (left) ring of quotients of 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.