Dense ideals and maximal quotient rings of incidence algebras

Date of Completion

January 2001






Let I(X, R) be an incidence algebra where X is a locally finite partially ordered set and R is a commutative ring with identity. Our intention is to construct the maximal or Utumi ring of quotients, which is defined for any ring T, of an incidence algebra I(X, R). Following the alternate description given by Findlay-Lambek, we use the dense ideals of the ring to construct its maximal ring of quotients. Since, in general, it is hard to determine all the dense ideals of a ring, instead we construct a basis of dense ideals that form the Gabriel topology on the dense ideals. Our starting point will be the fact that any dense ideal is also an essential ideal. In this research, results about essential ideals are used to obtain description of the dense ideals of an incidence algebra. Also, the necessary and sufficient conditions for I(X, R) to have a minimal dense ideal are stated. Further, this result is used to compute the maximal quotient ring of some incidence algebras. ^