UC Berkeley

The goal of this dissertation is to provide noncommutative generalizations of the following theorems from commutative algebra: (Cohen's Theorem) every ideal of a commutative ring R is finitely generated if and only if every prime ideal of R is finitely generated, and (Kaplansky's Theorems) every ideal of R is principal if and only if every prime ideal of R is principal, if and only if R is noetherian and every maximal ideal of R is principal. We approach this problem by introducing 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. 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.

After developing the theory of right Oka families, we proceed to the generalizations of the theorems stated above. Define a right ideal P of a ring R to be cocritical if the module R/P has larger Krull dimension than each of its proper factors. We prove that a ring is right noetherian (resp. a principal right ideal ring) if and only if all of its (essential) cocritical right ideals are finitely generated (resp. principal). We apply our methods to prove that a (left and right) noetherian ring is a principal right ideal ring if and only if all of its maximal right ideals are principal. Examples are provided to show that the left noetherian hypothesis cannot be omitted.

Finally, we compare these results with previous generalizations of these theorems, and are able to recover most of these with our methods.