Geodesic
On the Krull–Schmidt theorem for Artinian modules
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 16, Tome 263 (2000), pp. 187-192 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
@article{ZNSL_2000_263_a11,
     author = {K. I. Pimenov},
     title = {On the {Krull{\textendash}Schmidt} theorem for {Artinian} modules},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {187--192},
     year = {2000},
     volume = {263},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2000_263_a11/}
}
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K. I. Pimenov. On the Krull–Schmidt theorem for Artinian modules. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 16, Tome 263 (2000), pp. 187-192. http://geodesic.mathdoc.fr/item/ZNSL_2000_263_a11/