2012/02/02 by Mahmood Behboodi, Behboodi, Mahmood, Gholamreza Behboodi Eskandari +1
Mathematics · Neuroscience · #16D10 #16D70 #16P20 (Primary) 16N60 (Secondary) #Axon Guidance and Neuronal Signaling #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Rings and Algebras (math.RA) #Rings, Modules, and Algebras #math.AC #math.RA #msc:16D10 #msc:16D70 #msc:16N60 #msc:16P20
paper · pdf · doi:10.48550/arxiv.1202.0386
10 Pages
openalex publication_date 2012/02/02 · arxiv created 2012/10/15 · arxiv updated 2012/10/16 · openalex created_date 2022/09/04 · openalex updated_date 2026/07/28
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 FGC-ring if and only if R is a right FGC-ring, if and only if, R is a principal ideal ring. Moreover, we obtain that if R=Πi=1n Ri is a finite product of Noetherian duo-rings Ri where each Ri is prime or local, then R is a left FGC-ring if and only if R is a principal ideal ring.each Ri is prime or local, then R is a left FGC-ring if and only if R is a principal ideal ring.