Multiplication modules with Krull dimension

Abstract

In ring theory, it is shown that a commutative ring R with Krull dimension has classical Krull dimension and satisfies k.dim(R)=cl.k.dim(R). Moreover, R has only a finite number of distinct minimal prime ideals and some finite product of the minimal primes is zero (see Gordon and Robson [9, Theorem 8.12, Corollary 8.14, and Proposition 7.3]). In this paper, we give a generalization of these facts for multiplication modules over commutative rings. Actually, among other results, we prove that if M is a multiplication R-module with Krull dimension, then: (i) M is finitely generated, (ii) R has finitely many minimal prime ideals P_1, ..., P_n of Ann(M) such that P_1^k...P_n^kM=(0) for some k \geq 1, and (iii) M has classical Krull dimension and k.dim(M)=cl.k.dim(M)=k.dim(M/PM)= cl.k.dim(M/PM) for some prime ideal P of R.