Multiplication modules with Krull dimension

Maryam Molakarimi 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 P1,...,Pn of Ann(M) such that Pk1...PknM = (0) for some k ≥ 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. Key Words: Krull dimension, classical Krull dimension, multiplication module, prime submodul