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In this paper we study (non-commutative) rings $R$ over which every finitelygenerated left module is a direct sum of cyclic modules (called leftFGC-rings). The commutative case was a well-known problem studied and solved in1970s by various authors. It is shown that a Noetherian local left FGC-ring iseither an Artinian principal left ideal ring, or an Artinian principal rightideal ring, or a prime ring over which every two-sided ideal is principal as aleft and a right ideal. In particular, it is shown that a Noetherian localduo-ring $R$ is a left FGC-ring if and only if $R$ is a right FGC-ring, if andonly if, $R$ is a principal ideal ring. Moreover, we obtain that if$R=\Pi_{i=1}^n R_i$ is a finite product of Noetherian duo-rings $R_i$ whereeach $R_i$ is prime or local, then $R$ is a left FGC-ring if and only if $R$ isa principal ideal ring.each $R_i$ is prime or local, then $R$ is a leftFGC-ring if and only if $R$ is a principal ideal ring.

On rings each of whose finitely generated modules is a direct sum of cyclic modules