On the Generalized Quotient Rings and Their Reality

The purpose of this paper is to investigate the so-called multiplicatively directed families of ideals and their generalized quotient rings.This paper is divided into two sections.In the first section,We give the structure of the generalized quotient ring for a multiplicatively directed family of ideals,and obtain some basic property of the quotient ring.In section 2,We establish some conditions for a generalized quotient ring to be real.