U-factorization of ideals

Abstract

We study the factorization of ideals of a commutative ring, in the context of the U-factorization framework introduced by Fletcher. This leads to several “U-factorability” properties weaker than unique U-factorization. We characterize these notions, determine the implications between them, and give several examples to illustrate the differences. For example, we show that a ring is a general ZPI-ring if and only if its monoid of ideals has unique factorization in the sense of Fletcher. We also examine how these “U-factorability” properties behave with respect to several ring-theoretic constructions.

Keywords:

• Commutative rings
• factorization
• ideals
• multiplicative ideal theory
• U-factorization

2010 Mathematics Subject Classification:

• 13A05
• 13A15
• 13E99
• 13F15

Acknowledgements

We would like to express our gratitude to the referee(s) for the thorough reading and suggestions that have improved the quality of the paper.