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arXiv · 1709.02748

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

Danny A. J. Gomez-Ramirez·Juan D. Velez·Edisson Gallego
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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.

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Danny A. J. Gomez-Ramirez, Juan D. Velez, Edisson Gallego. 2017-09-08. On Preservation Properties and a Special Algebraic Characterization of Some Stronger Forms of the Noetherian Condition. https://arxiv.org/abs/1709.02748

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