27 - M. Behboodi (with M. Mollakarimi), Multiplication modules with Krull dimension, Turk. J. Math., 36, (2012), 550-559. (ISI)
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_k1 . . .P_knM =(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.