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On the Maximal Spectrum of Semiprimitive Multiplication Modules

 Printed: Sep 2008
  • Karim Samei
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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 one-point 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.
Keywords: multiplication module, semiprimitive module, Gelfand module, Zariski topolog multiplication module, semiprimitive module, Gelfand module, Zariski topolog
MSC Classifications: 13C13 show english descriptions Other special types 13C13 - Other special types

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