On the Krull–Schmidt theorem for Artinian modules

A property of rings generalizing commutativity is introduced. If a ring satisfies this property, then the Krull-Schmidt theorem holds for Artinian modules over the ring. In particular, this property is fulfilled for local rings of finite rank and for rings such that their centers are surjectively mapped by the natural projection onto the factor with respect to the radical of the ring. A local ring for which the property fails is constructed; for the direct decompositions of Artinian modules over this ring there appear anomalies similar to the anomalies of direct decompositions of torsion-free Abelian groups of finite rank.