Dense Ideals and Maximal Quotient Rings of Incidence Algebras

Abstract

In this paper, we determine a core subset of dense ideals and left dense ideals of some incidence algebras. As an application and the motivation of this work, we compute the maximal left quotient ring of some incidence algebras. In a ring T, if a minimal left dense ideal, D exists, then the maximal left quotient ring of T is isomorphic to the ring of T-module homomorphisms of D into D (Lambek, J. (1963). On Utumi's ring of quotients. Can. J. Math. 15:363–370). Hence, in the case of an incidence algebra, we give necessary and sufficient conditions for a minimal left dense ideal to exist and give a description of this ideal when it exists.

Key Words:

• Incidence algebras
• Quotient rings
• Dense ideals

Acknowledgments

This paper is a result of the author's PhD thesis at the University of Connecticut. The author wishes to thank Spiegel for his guidance as an advisor. Also, the author would like to thank the referee for his/her many helpful suggestions to make the paper more reader friendly.