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Primes Associated to an Ideal
Stephen McAdam

Contemporary Mathematics
1989; 167 pp; softcover
Volume: 102
ISBN-10: 0-8218-5108-X
ISBN-13: 978-0-8218-5108-1
List Price: US$48
Member Price: US$38.40
Order Code: CONM/102
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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.

Table of Contents

  • Contents
  • Basic results
  • Examples
  • Essential and asymptotic sequences
  • Schenzel's theorems
  • The relative Rees ring of \(I\) and \(J\)
  • Two asymptotic functions
  • Finite transforms
  • Essential primes and projective extensions
  • Persistent primes and projective extensions
  • Prime divisors of principal ideals
  • Sporadic primes
  • Irrelevant prime divisors of \(uR(I)\)
  • Generalizations to many ideals
  • Grad functions
  • Partial orderings on grade functions
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