ZARISKI-LIKE TOPOLOGY ON THE CLASSICALPRIME SPECTRUM OF A MODULEt

Let R be a commutative ring with identity and let Mbe an R-module. A proper submodule P of AI is called a classical prime submodule if abm E P for a., b E R, and m E n, impliesthat am E P or bm E P. The classical prime spectrum CI.Spec(M)is defined to be the set of all classical prime sub modules of AI.The aim of this paper is to introduce and study a topology onCI.Spec(M), which generalizes the Zariski topology of R to M,called Zariski-like topology of AI. In particular, we investigate this topological space from the point of view of spectral spaces. It is shown that if M is a Noetherian (or an Artinian) R-module, thenCI.Spec(M) with the Zariski-like topology is a spectral space, i.e.,there exists a commutative ring S such that CLSpec(M) with the Zariski-like topology is homeomorphic to Spec(S) with the usual Zariski topology.