Multiplication modules with Krull dimension

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 P1k...PnkM=(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.

Anahtar Kelimeler:

Krull dimension, classical Krull dimension, multiplication module, prime submodule

Multiplication modules with Krull dimension

Keywords:

Krull dimension, classical Krull dimension, multiplication module, prime submodule,

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Mahmood BEHBOODI1,2, Maryam MOLAKARIMI1 Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, 83111, IRAN e-mails: mbehbood@cc.iut.ac.ir, m.molakarimi@math.iut.ac.ir
School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, 19395-5746, IRAN