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| Paper IPM / M / 9552 |
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| Abstract: | |
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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 1 of R. As defined for a commutative ring R, an R-module M is said to be semiprimitive if the intersection of maximal sub modules 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 sub modules of M is an essential submodule if and only if Max(M) is an almost discrete space. The set of uniform sub modules of M and the set of minimal sub modules 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 ifMax(M) is almost discrete.
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