On Preservation Properties and a Special Algebraic Characterization of Some Stronger Forms of the Noetherian Condition

TL;DR

This paper provides elementary proofs for preservation of the Noetherian property under certain conditions, explores algebraic characterizations of Noetherian rings, and discusses the necessity of completion conditions.

Contribution

It introduces new elementary proofs for Noetherian preservation and offers a characterization of decomposable Noetherian rings, highlighting the importance of completion conditions.

Findings

Proof that Noetherian property is preserved under specific ideal and completion conditions

Elementary proof that formal power series rings over Noetherian rings are Noetherian

Counterexample showing the necessity of the completion condition

Abstract

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.