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.