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Paper   IPM / M / 17341
School of Mathematics
  Title:   Commutative rings whose proper ideals are pure-semisimple
1.  Mahmood Behboodi
2.  Ali Moradzadeh-Dehkordi (Joint with S. Baghdari)
  Status:   Published
  Journal: Comm. Algebra
  Year:  2023
  Pages:   DOI: 10.1080/00927872.2023.2217720, 1-9
  Supported by:  IPM
Recall that an R-module M is pure-semisimple every module in the category σ[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 (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 M= Rx⊕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.

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