Unique factorization of ideals into nonfactorable ideals

The purpose of this note is to prove a theorem which shows a connection between the definition of a prime ideal in classical algebraic number theory and the usual definition of a prime ideal. A proper ideal in an integral domain with unit element is an ideal different from the unit ideal and the zero ideal. An ideal A will be called nonfactorable provided A is a proper ideal and A = BC (where B and C are ideals) implies that either B or C is the unit ideal.