Multiplication modules with Krull dimension

Authors : Mahmood BEHBOODI, Maryam MOLAKARIMI

Pages : 550-559

View : 9 | Download : 6

Publication Date : 0000-00-00

Article Type : Research Paper

Abstract :In ring theory, it is shown that a commutative ring R with Krull dimension has classical Krull dimension and satisfies k.diminsert ignore into journalissuearticles values(R);=cl.k.diminsert ignore into journalissuearticles values(R);. Moreover, R has only a finite number of distinct minimal prime ideals and some finite product of the minimal primes is zero insert ignore into journalissuearticles values(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: insert ignore into journalissuearticles values(i); M is finitely generated, insert ignore into journalissuearticles values(ii); R has finitely many minimal prime ideals P1, ..., Pn of Anninsert ignore into journalissuearticles values(M); such that P1k...PnkM=insert ignore into journalissuearticles values(0); for some k \geq 1, and insert ignore into journalissuearticles values(iii); M has classical Krull dimension and k.diminsert ignore into journalissuearticles values(M);=cl.k.diminsert ignore into journalissuearticles values(M);=k.diminsert ignore into journalissuearticles values(M/PM);= cl.k.diminsert ignore into journalissuearticles values(M/PM); for some prime ideal P of R.
Keywords : Krull dimension, classical Krull dimension, multiplication module, prime submodule