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| Paper IPM / M / 8112 |
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| Abstract: | |
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In this article, we introduce and study a generalization of the
classical krull dimension for a module RM. 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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