Unique decompositions into regular ideals for Marot rings

Abstract

Let R be a commutative ring. We say that R has the unique decomposition into regular ideals (UDRI) property if, for any R-module which decomposes into a finite direct sum of regular ideals, this decomposition is unique up to the order and isomorphism class of the regular ideals. In this paper, we will prove some preliminary results for Marot rings whose regular ideals are finitely generated and give a necessary and sufficient condition for these rings to satisfy the UDRI property.

Keywords:

• h-local rings
• Marot rings
• regular localization
• valuation rings

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 13F30
• 13A18
• 13B30
• 13F99

Acknowledgment

The authors would like to thank the anonymous referee for his/her careful reading and suggestions.