14 - M. Behboodi (with S. H. Shojaee), On the classical krull dimension for modules, Int. J. Algebra, 3 (6) (2009), 287 - 296.
Let R be a ring and let M be a left R-module. The classical Krull dimension of M was defined to be the supremum of the lengths of all strong chains of prime submodules of M and denoted by cl.k.dim(M). In this article, we show that for every finitely generated faithful R-module M, cl.k.dim(M) exists if and only if cl.k.dim(R) exists and cl.k.dim(M)=cl.k.dim(R) in case one of them exists. This yields for every finitely generated moduleM over a commutative ring R, cl.k.dim(M) = cl.k.dim(R/I) where I = Ann(M), in case one of them exists. We show that, in general, this is not true, in the non-commutative case. Also, it is shown that every semisimple left R-module has classical Krull dimension zero. Furthermore, modules with classical Krull dimension -1 are also studied.