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- W4379513505 abstract "Recall that an R-module M is pure-semisimple if 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 (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 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." @default.
- W4379513505 created "2023-06-07" @default.
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- W4379513505 date "2023-06-06" @default.
- W4379513505 modified "2023-10-15" @default.
- W4379513505 title "Commutative rings whose proper ideals are pure-semisimple" @default.
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- W4379513505 doi "https://doi.org/10.1080/00927872.2023.2217720" @default.
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