Intermediate normalizing extensions

Abstract:

Relationships between the prime ideals of a ring $R$ and of a normalizing extension $S$ have been studied by several authors recently. In this work, most of these known results are extended to give relationships between the prime ideals of $R$ and of $T$ where $T$ is a ring with $R \subset T \subset S$, and $S$ is a normalizing extension of $R$: such rings $T$ are called intermediate normalizing extensions of $R$. One result ("Cutting Down") asserts that for any prime ideal $J$ of $T$, $J \cap R$ is the intersection of a finite set of prime ideals ${P_i}$ of $R$, uniquely defined by $J$, whose corresponding factor rings $R/{P_i}$ are mutually isomorphic. The minimal members of the family of ${P_i}$’s are the primes of $R$ minimal over $J \cap R$, and an "incomparability" theorem is proved which shows that no two comparable primes of $T$ can have their intersections with $R$ share a common minimal prime. Other results include versions of the "lying over" and "going up" theorems, proofs that chain conditions such as right Goldie or right Noetherian pass between $T/J$ and each of the rings $R/{P_i}$, and a demonstration that the "additivity principle" holds.