CLASSICAL ZARISKI TOPOLOGY OF MODULES AND SPECTRAL SPACES II

Year 2008, Volume: 4 Issue: 4, 131 - 148, 01.12.2008

M. Behboodi M. R. Haddadi

Abstract

In this paper we continue our study of classical Zariski topology of modules, that was introduced in Part I (see [2]). For a left R-module M, the prime spectrum Spec(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, Spec(RM) with the patch topology is Hausdorf and it is disconnected provided |Spec(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 (ACC) on intersection of prime submodules, then Spec(RM) is a spectral space, i.e., Spec(RM) is homeomorphic to Spec(S) for some commutative ring S. This yields if M is a Noetherian left R-module or R is a PI-ring (or an FBN-ring) and M is an Artinian left R-module, then Spec(RM) is a spectral space. Finally, we show that for every Noetherian left R-module M, Max(M) (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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• M. Behboodi and M.R. Haddadi Department of Mathematical Science, Isfahan University of Technology, 83111 Isfahan, Iran e-mails: [e-mail] (M. Behboodi) [e-mail] (M.R. Haddadi)