A (one-sided) Prime Ideal Principle for Noncommutative Rings

In this paper we study certain families of right ideals in noncommutative rings, called right Oka families, generalizing previous work on commutative rings by T.Y. Lam and the author. As in the commutative case, we prove that the right Oka families in a ring R correspond bijectively to the classes of cyclic right R-modules that are closed under extensions. We define completely prime right ideals and prove the Completely Prime Ideal Principle, which states that a right ideal maximal in the complement of a right Oka family is completely prime. We exploit the connection with cyclic modules to provide many examples of right Oka families. We also show that the well-studied right Gabriel filters are examples of divisible right Oka families, which enjoy a stronger version of the Completely Prime Ideal Principle. Our methods produce some new results that generalize well-known facts from commutative algebra, and they also recover earlier theorems stating that certain noncommutative rings are domains—namely, proper right PCI rings and rings with the right restricted minimum condition that are not right artinian.