Elementary preservation of the Noetherian condition and the potential applications in computational physics

Abstract

Abstract We give an elementary proof 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. Furthermore, we discuss the potential applications that this new elementary proof possesses regarding the simplification of conceptual generations in mathematics and computational physics based on the new computational paradigm of Artificial Mathematical Intelligence. 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.