An Intersection Condition for Prime Ideals

In this chapter, all rings are assumed to be commutative and to contain a unity element. A widely used result in commutative ring theory is the Prime Avoidance Theorem, which states that an ideal contained in a finite union of prime ideals is contained in one of the prime ideals. The chapter provides a proposition that gives equivalent conditions on a fixed prime ideal in order that the prime ideal contains the intersection of a family of prime ideals only if the prime ideal contains one of the prime ideals of the family. It also presents basic local results concerning the intersection condition.