Multiplication modules with Krull dimension

作者：Behboodi Mahmood^*; Molakarimi Maryam

来源：Turkish Journal of Mathematics, 2012, 36(4): 550-559.

DOI：10.3906/mat-1101-63

• Krull dimension
• classical Krull dimension
• multiplication module
• prime submodule

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 Cordon 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 It-module with Krull dimension, then: (i) 111 is finitely generated, (ii) R has finitely many minimal prime ideals P-1, ... ,P-n of Ann (M) such that P-1(k) ... (PnM)-M-k = (0) for some k %26gt;= 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 sonic prime ideal P of R.

• 出版日期2012