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 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 S f 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 , f g , and r f in I * ( X , R ) such that ( f + g ) ( x + y ) = f ( x , y ) + g ( x , y ) , f g ( x , y ) = ∑ x ≤ z ≤ y f ( x , z ) g ( z , y ) , r f ( 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.