Various Notions of Associated Prime Ideals

In the theory of modules over commutative rings there are several possibilities of defining associated prime ideals. The usual definition of an associated prime ideal p for a module M is that p is the annihilator of an element of M . In [2] §1 exercise 17 a generalization of this notion is given. p is called weakly associated (faiblement associé) to M if p is minimal in the set of the prime ideals containing the annihilator of an element of M (see Definition 2.13). In this paper a further generalization of this notion will be given (Definition 2.1). We use ideas of Krull [3]. As long as the modules are noetherian all these definitions are equivalent. But for non noetherian R-modules this is no longer true, even if the ring R is noetherian. In this paper we give a selfcontained introduction to the various concepts and discuss their relation with the support and the radical of a module. Then we illustrate by examples the scope of the notions. For a comprehensive introduction the theory we refer to the now classic lecture notes [7] of Serre and to [2]. Another extensive exposition of the general theory with many examples was given by Stefan Mittelbach in [5]. Throughout this paper “ring” always denotes a commutative ring with unit element denoted by 1. If M is an R-module we always assume that 1 · x = x for all x ∈M . In the first section we recall some basic definitions and facts from “additive ideal theory”.