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21 - M. Behboodi (with A.Ghorbani, A.Moradzadeh-Dehkordi), Commutative Noetherian local rings whose ideals are direct sums of cyclic modules, J. Algebra, 345 (2011) 257–265. (ISI)

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.
September, 2011


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