UNIQUE FACTORIZATION AND BIRTH OF ALMOST PRIMES

ABSTRACT

In studying unique factorization of domains we encountered a property of ideals. Using that we define the notion of almost prime ideals and prove that in Noetherian domains almost prime ideals are primary. We also prove that in a regular domain almost primes are precisely primes. Further, we define strictly nonprime ideals and study some inter relations between almost prime ideals, strictly nonprime ideals and factorization of ideals.

Key Words:

• Almost prime ideal
• Prime ideal
• Regular ring
• Strictly nonprime ideal
• Unique factorization domain

ACKNOWLEDGMENTS

The second author expresses his thanks to Moshe Roitman for several useful e-mail exchanges and also to Stephen McAdam for a lot of valuable suggestions on the first draft of the manuscript. Theorem 2.11 IS essentially due to him. The authors thank the referee for his careful reading of the manuscript and suggesting improvements in the presentation.

Notes

^#Communicated by I. Swanson.