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.
📌 3 Key Takeaways
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Objective & Hypothesis: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).
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Key Findings & Data:The commutative case was a well-known problem studied and solved in 1970s by various authors.
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Conclusion & Impact: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.
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-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.
BEHBOODI, M., ESKANDARI, G. B. (2022). On rings over which every finitely generated module is a direct sum of cyclic modules. Hacettepe Journal of Mathematics and Statistics, Cilt: 45 - Sayı: 5, 1335–1342. https://ajindex.com/articles/530113