Ideal factoriality, semistar operations, and quasiprincipal Ideals

Abstract

The w-operation, a “smoother” variant of the classic t-operation on the set of ideals of a commutative ring R, has recently attracted significant attention. We study when R’s monoid of w-ideals is “factorial” in some sense. For example, we show that every proper w-ideal of R has a unique up to order irredundant w-factorization into w-unfactorable w-ideals if and only if R is a finite direct product of Krull domains and principal ideal rings, if and only if every w-ideal is w-quasiprincipal. We prove several analogous results regarding other kinds of “w-ideal factoriality” and generalize these results via semistar operations.

KEYWORDS:

• Semistar ideal factoriality
• semistar Nagata ring
• semistar π-rings/general ZPI-rings
• semistar quasiprincipal ideal
• w-operation

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 13F05
• 13A15
• 13F15
• 13A05
• 13F99

Acknowledgments

We would like to thank the anonymous referee for taking the time to carefully read our article and provide several thoughtful comments, suggestions, and corrections. This helpful feedback resulted in a more clear and complete article.