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In this article, we introduce and study a generalization of the
classical krull dimension for a module $_{R}M$. 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 Cohen's Theorem for multiplication modules.
Also, PI-rings whose nonzero modules have zero classical Krull
dimension are characterized.
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