Modules with Noetherian Spectrum

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 dimension
• Prime spectrum Noetherian spectrum
• Prime submodule
• Zariski topology

2000 Mathematics Subject Classification:

• Primary 13C99
• Secondary 13A10, 13C13, 13C15

ACKNOWLEDGMENT

The author thanks the referee for helpful and valuable suggestions and comments.

Notes

Communicated by I. Swanson.