2017/09/08 by Danny A. J. Gómez–Ramírez, Gomez-Ramirez, Danny A. J., Juan D. Vélez +3
Mathematics · #13E05 #Advanced Topics in Algebra #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.1709.02748
openalex publication_date 2017/09/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We give an elementary proof prove of the preservation of the Noetherian condition for commutative rings with unity R having at least one finitely generated ideal I such that the quotient ring is again finitely generated, and R is I-adically complete. Moreover, we offer as a direct corollary a new elementary proof of the fact that if a ring is Noetherian then the corresponding ring of formal power series in finitely many variables is Noetherian. In addition, we give a counterexample showing that the `completion' condition cannot be avoided on the former theorem. Lastly, we give an elementary characterization of Noetherian commutative rings that can be decomposed as a finite direct product of fields.