CLASSICAL ZARISKI TOPOLOGY OF MODULES AND SPECTRAL SPACES II

Authors : M BEHBOODİ, M R HADDADİ

Pages : 131-148

View : 35 | Download : 10

Publication Date : 2008-12-01

Article Type : Research Paper

Abstract :In this paper we continue our study of classical Zariski topology of modules, that was introduced in Part I insert ignore into journalissuearticles values(see [2]);. For a left R-module M, the prime spectrum Specinsert ignore into journalissuearticles values(RM); of M is the collection of all prime submodules. First, we study some continuous mappings which are induced from some natural homomorphisms. Then we generalize the patch topology of rings to modules, and show that for every left R-module M, Specinsert ignore into journalissuearticles values(RM); with the patch topology is Hausdorf and it is disconnected provided |Specinsert ignore into journalissuearticles values(RM);| > 1. Next, by applying Hochster’s characterization of a spectral space, we show that if M is a left R-module such that M has ascending chain condition insert ignore into journalissuearticles values(ACC); on intersection of prime submodules, then Specinsert ignore into journalissuearticles values(RM); is a spectral space, i.e., Specinsert ignore into journalissuearticles values(RM); is homeomorphic to Specinsert ignore into journalissuearticles values(S); for some commutative ring S. This yields if M is a Noetherian left R-module or R is a PI-ring insert ignore into journalissuearticles values(or an FBN-ring); and M is an Artinian left R-module, then Specinsert ignore into journalissuearticles values(RM); is a spectral space. Finally, we show that for every Noetherian left R-module M, Maxinsert ignore into journalissuearticles values(M); insert ignore into journalissuearticles values(with the classical Zariski topology); is homeomorphic with the maximal ideal space of some commutative ring S.
Keywords : Prime submodule, Prime spectrum, Classical Zariski topology, Spectral space, Patch topology

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