This book discusses five closely related sets of prime ideals associated to an ideal \(I\) in a Noetherian ring: the persistent, asymptotic, quintasymptotic, essential, and quintessential primes of \(I\). Since the appearance of the author's last book on this subject, which focused on the first two of these prime ideals, the other three sets were developed and new results were obtained for the first two. Current results are scattered over some three dozen papers, making it difficult for interested readers to become familiar with the subject.

The aim of this book is to present in an efficient way the most important and interesting ideas in the subject and to show how these prime ideals reveal information about both \(I\) and the ring. Because the required background consists of little more than a standard one-year course in commutative ring theory, the book should be acccessible to graduate students. The work is primarily intended for commutative ring theorists, but noncommutative ring theorists and algebraic geometers may also find it of interest.