  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.
