CLASSICAL ZARISKI TOPOLOGY OF MODULES AND SPECTRAL SPACES II

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

References

1. M. Behboodi, A generalization of the classical krull dimension for modules, J. Algebra, 305(2006), 1128-1148.
2. M. Behboodi and M. R. Haddadi, Classical Zariski topology of modules and spectral spaces I, Int. Electron. J. Algebra, 4 (2008), 104-130.
3. N. Bourbaki, Algebra Commutative, Chap, 1.2, Hermann, Paris, 1961.
4. K. R. Goodearl and R. B. Warfield, An Introduction to Non-commutative Noetherian Rings (Second Edition), London Math. Soc. Student Texts 16, Cambridge University Press, Cambridge, 2004.
5. M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc., 137 (1969), 43-60.
6. C. P. Lu, Prime submodule of modules, Math. Univ. Sancti. Pauli., 33 (1984), 69.
7. J. R. Mankres, Topology, A first course, Prentice-Hall, Inc. Eaglewood Cliffs, New Jersey, 1975.
8. R. L. McCasland and M. E. Moore, Prime submodules, Comm. Algebra, 20 (1992), 1803-1817.

9. R. L. McCasland, M. E. Moore and P. F. Smith, On the spectrum of a module over a commutative ring, Comm. Algebra, 25 (1997), 79-103.
10. R. L. McCasland and P. F. Smith, Prime submodules of Noetherian modules, Rocky Mountain J. Math., 23 (1993), 1041-1062.
11. J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, Wily, Chichester, 1987.
12. 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)