1. Commutative Noetherian local rings whose ideals are direct sums of cyclic modules
- Author
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Behboodi, M., Ghorbani, A., and Moradzadeh-Dehkordi, A.
- Subjects
- *
COMMUTATIVE rings , *NOETHERIAN rings , *LOCAL rings (Algebra) , *IDEALS (Algebra) , *MODULES (Algebra) , *COMMUTATIVE algebra , *ARTIN rings , *MATHEMATICS - Abstract
Abstract: A theorem from commutative algebra due to Köthe and Cohen-Kaplansky states that, “a commutative ring R has the property that every R-module is a direct sum of cyclic modules if and only if R is an Artinian principal ideal ring”. Therefore, an interesting natural question of this sort is “whether the same is true if one only assumes that every ideal is a direct sum of cyclic modules?” The goal of this paper is to answer this question in the case R is a finite direct product of commutative Noetherian local rings. The structure of such rings is completely described. In particular, this yields characterizations of all commutative Artinian rings with this property. [Copyright &y& Elsevier]
- Published
- 2011
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