Mathematics – Rings and Algebras
Scientific paper
2012-02-02
Mathematics
Rings and Algebras
12 Pages
Scientific paper
In this paper we study (non-commutative) rings $R$ over which every finitely generated left module is a direct sum of cyclic modules (called left FGC-rings). The commutative case was a well-lnown problem studied and solved in 1970s by various authors. The main result of this paper shows 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. As a consequence, we obtain that if $R=\Pi_{i=1}^n R_i$ is a finite product of Noetherian duo-rings $R_i$ where each $R_i$ is prime or local, then $R$ is a left FGC-ring if and only if $R$ is a principal ideal ring.
Behboodi Mahmood
Eskandari Gholamreza Behboodi
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