NOTHERIAN LOCAL RINGS WHOSE IDEALS ARE DIRECT SUM OF PRINCIPAL IDEALS

It was shown by K¨othe and Cohen-Kaplansky that “a commutative ring R has the property that every module is a direct sum of cyclic modules if and only if R is an Artinian principal ideal ring”. Therefore, an important 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 determine all commutative Artinian rings with this property.