Strong Krull primes and flat modules

TL;DR

This paper introduces strong Krull primes as a suitable replacement for associated primes over non-Noetherian rings, establishing their role in characterizing flat modules and generalizing classical theorems.

Contribution

It demonstrates that strong Krull primes closely mimic associated primes over Noetherian rings and extends key flatness theorems using these primes, with new results and counterexamples.

Findings

Strong Krull primes behave like associated primes over Noetherian rings.

An analogue of Epstein and Yao's theorem is proved using strong Krull primes.

Containment and equality results for flat base change are established with strong Krull primes.

Abstract

There are several theorems describing the intricate relationship between flatness and associated primes over commutative Noetherian rings. However, associated primes are known to act badly over non-Noetherian rings, so one needs a suitable replacement. In this paper, we show that the behavior of strong Krull primes most closely resembles that of associated primes over a Noetherian ring. We prove an analogue of a theorem of Epstein and Yao characterizing flat modules in terms of associated primes by replacing them with strong Krull primes. Also, we partly generalize a classical equational theorem regarding flat base change and associated primes in Noetherian rings. That is, when associated primes are replaced by strong Krull primes, we show containment in general and equality in many special cases. One application is of interest over any Noetherian ring of prime characteristic. We also give numerous examples to show that our results fail if other popular generalizations of associated primes are used in place of strong Krull primes.