Maximal Ideal Transforms of Noetherian Rings

Abstract

Let $R$ be a commutative Noetherian ring with unit. Let $T$ be the set of all elements of the total quotient ring of $R$ whose conductor to $R$ contains a power of a finite product of maximal ideals of $R$. If $A$ is any ring such that $R \subset A \subset T$, then $A/xA$ is a finite $R$ module for any non-zero-divisor $x$ in $R$. It follows that if, in addition, $R$ has no nonzero nilpotent elements, then any ring $A$ such that $R \subset A \subset T$ is Noetherian.