A generalization of the classical Krull dimension for modules

In this article, we introduce and study a generalization of the classical Krull dimension for a module . This is defined to be the length of the longest strong chain of prime submodules of M (defined later) and, denoted by Cl.K.dim(M). This notion is analogous to that of the usual classical Krull dimension of a ring. This dimension, Cl.K.dim(M) exists if and only if M has virtual acc on prime submodules; see Section 2. If R is a ring for which Cl.K.dim(R) exists, then for any left R-module M, Cl.K.dim(M) exists and is no larger than Cl.K.dim(R). Over any ring, all homogeneous semisimple modules and over a PI-ring (or an FBN-ring), all semisimple modules as well as, all Artinian modules with a prime submodule lie in the class of modules with classical Krull dimension zero. For a multiplication module over a commutative ring, the notion of classical Krull dimension and the usual prime dimension coincide. This yields that for a multiplication module M, Cl.K.dim(M) exists if and only if M has acc on prime submodules. As an application, we obtain a nice generalization of Cohenʹs Theorem for multiplication modules. Also, PI-rings whose nonzero modules have zero classical Krull dimension are characterized.