Modules with Noetherian Spectrum

Abstract

Abstract Let M be a module over a commutative ring R. A submodule P of M is called prime if P ≠ M and, whenever r ∈ R, e ∈ M, and re ∈ P, we have rM ⊆ P or e ∈ P. We let Spec(M) denote the set of all prime submodules of M. Using a topology analogous to the Zariski topology for Spec(R), we establish necessary and sufficient conditions for Spec(M) to be a Noetherian space. We produce some examples of modules with Noetherian spectrum that have not appeared in the literature previously. In particular, Laskerian modules and faithfully flat modules over Laskerian rings have Noetherian spectra. (The term Laskerian is defined in Section 3.) Key Words: Combinatorial dimensionPrime spectrum Noetherian spectrumPrime submoduleZariski topology2000 Mathematics Subject Classification: Primary 13C99Secondary 13A10, 13C13, 13C15 ACKNOWLEDGMENT The author thanks the referee for helpful and valuable suggestions and comments. Notes Communicated by I. Swanson.