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On the Maximal Spectrum of Semiprimitive Multiplication Modules


Published:20080901
Printed: Sep 2008
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Abstract
An $R$module $M$ is called a multiplication module if for each
submodule $N$ of $M$, $N=IM$ for some ideal $I$ of $R$. As
defined for a commutative ring $R$, an $R$module $M$ is said to be
semiprimitive if the intersection of maximal submodules of $M$ is
zero. The maximal spectra of a semiprimitive multiplication
module $M$ are studied. The isolated points of $\Max(M)$ are
characterized algebraically. The relationships among the maximal
spectra of $M$, $\Soc(M)$ and $\Ass(M)$ are studied. It is shown
that $\Soc(M)$ is exactly the set of all elements of $M$ which
belongs to every maximal submodule of $M$ except for a finite
number. If $\Max(M)$ is infinite, $\Max(M)$ is a onepoint
compactification of a discrete space if and only if $M$ is Gelfand and for
some maximal submodule $K$, $\Soc(M)$ is the intersection of all
prime submodules of $M$ contained in $K$. When $M$ is a
semiprimitive Gelfand module, we prove that every intersection
of essential submodules of $M$ is an essential submodule if and only if
$\Max(M)$ is an almost discrete space. The set of uniform
submodules of $M$ and the set of minimal submodules of $M$
coincide. $\Ann(\Soc(M))M$ is a summand submodule of $M$ if and only if
$\Max(M)$ is the union of two disjoint open subspaces $A$ and
$N$, where $A$ is almost discrete and $N$ is dense in itself. In
particular, $\Ann(\Soc(M))=\Ann(M)$ if and only if $\Max(M)$ is almost
discrete.