Commutative rings whose proper ideals are pure-semisimple

Abstract

Recall that an R-module M is pure-semisimple if every module in the category $\sigma [M]$ is a direct sum of finitely generated (and indecomposable) modules. A theorem from commutative algebra due to Köthe, Cohen-Kaplansky and Griffith states that “a commutative ring R is pure-semisimple (i.e., every R-module is a direct sum of finitely generated modules) if and only if every R-module is a direct sum of cyclic modules, if and only if, R is an Artinian principal ideal ring”. Consequently, every (or finitely generated, cyclic) ideal of R is pure-semisimple if and only if R is an Artinian principal ideal ring. Therefore, a natural question of this sort is “whether the same is true if one only assumes that every proper ideal of R is pure-semisimple?” The goal of this paper is to answer this question. The structure of such rings is completely described as Artinian principal ideal rings or local rings R with the maximal ideals $\mathcal{M}=Rx\oplus T$ which Rx is Artinian uniserial and T is semisimple. Also, we give several characterizations for commutative rings whose proper principal (finitely generated) ideals are pure-semisimple.

Keywords:

• Artinian principal ideal ring
• direct sum decomposition
• pure-semisimple module
• serial module

2020 Mathematics Subject Classification:

• 13C05
• 13F10
• 13H99
• 16D70

Acknowledgments

The authors owe a great debt to the referee who has carefully read earlier versions of this paper and made significant suggestions for improvement. We would like to express our deep appreciation for the referee’s work.

Additional information

Funding

The research of the second and the third authors was in part supported by grants from IPM (No. 1401130213 and No. 1401160414). This research is partially carried out in the IPM-Isfahan Branch.