Maximal ideal transforms of Noetherian rings

Abstract

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