Locally noetherian commutative rings

Abstract:

This paper centers around the theorem that a commutative ring $R$ is noetherian if every ${R_P},P$ prime, is noetherian and every finitely generated ideal of $R$ has only finitely many weak-Bourbaki associated primes. A more precise local version of this theorem is also given, and examples are presented to show that the assumption on the weak-Bourbaki primes cannot be deleted nor replaced by the assumption that Spec $(R)$ is noetherian. Moreover, an alternative statement of the theorem using a variation of the weak-Bourbaki associated primes is investigated. The proof of the theorem involves a knowledge of the behavior of associated primes of an ideal under quotient ring extension, and the paper concludes with some remarks on this behavior in the more general setting of flat ring extensions.