On rings over which every finitely generated module is a direct sum of cyclic modules

Authors : M. BEHBOODİ, G. Behboodi ESKANDARİ

Pages : 1335-1342

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Publication Date : 2016-10-01

Article Type : Research Paper

Abstract :In this paper we study insert ignore into journalissuearticles values(non-commutative); rings R over which every finitely generated left module is a direct sum of cyclic modules insert ignore into journalissuearticles values(called left FGC-rings);. The commutative case was a well-known problem studied and solved in 1970s by various authors. It is shown that a Noetherian local left FGC-ring is either an Artinian principal left ideal ring, or an Artinian principal right ideal ring, or a prime ring over which every two-sided ideal is principal as a left and a right ideal. In particular, it is shown that a Noetherian local duo-ring R is a left FGCring if and only if R is a right FGC-ring, if and only if, R is a principal ideal ring.
Keywords : Cyclic modules, FGC rings, duo rings, principal ideal rings