** Authors:**
MAHMOOD BEHBOODI, MARYAM MOLAKARIMI

** 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.

** Keywords: **
Krull dimension, classical Krull dimension, multiplication
module, prime submodule

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