A scheme associated to modules over commutative rings

Abstract

For any module M over a commutative ring R with identity, we consider $\text{Sch}(M)$ as a certain class of prime submodules modulo the equivalence relation that two such prime submodules are the same if their colon ideals are equal. We topologize $\text{Sch}(M)$ and introduce a sheaf ${\mathcal{O}}_{\text{Sch}(M)}$ of commutative rings on it, which makes $(\text{Sch}(M),{\mathcal{O}}_{\text{Sch}(M)})$ into a scheme. In particular, if M is a module over a Noetherian ring R, then $(\text{Sch}(M),{\mathcal{O}}_{\text{Sch}(M)})$ is a locally Noetherian scheme. Among others, we give sufficient conditions for $\text{Sch}(M)$ to be an affine scheme.

Keywords:

• Affine scheme
• morphism of schemes
• prime submodule
• scheme
• sheaf

2020 Mathematics Subject Classification:

• Primary: 13C99
• Secondary: 14A15

Disclosure Statement

The authors declare that they have no conflict of interest, and also no funds, grants, or other support were received during the preparation of this manuscript.