On \tilde{Spec}(M) Topology of Module M over Commutative Rings

TL;DR

The paper introduces the $\widetilde{\operatorname{Spec}(M)}$ topology on the prime submodule spectrum of an $R$-module $M$, defined via closed sets $\widetilde{V}(S)$ and opens $\widetilde{D}(S)$ built from multiplicatively closed subsets of $R$. It proves that isolated points are precisely the minimal prime submodules and that, for multiplication modules, zero-dimensionality is equivalent to discreteness; it then develops the separation axioms, connectivity, nestedness, and Lindelöf/compactness theory, including an explicit Lindelöf-but-not-quasi-compact example. The work connects algebraic properties of $M$ and $R$ to topological features of $\widetilde{\operatorname{Spec}(M)}$, providing structural criteria for $T_0$, $T_1$, and $T_3$ spaces and analyzing how compactness relates to ring-theoretic localness (e.g., semi-locality) in the finitely generated faithful case. Finally, it gives constructive conditions for when $\widetilde{\operatorname{Spec}(M)}$ is compact or Lindelöf and illustrates these with examples, including cases where Lindelöfness fails or implies non-quasi-compactness.

Abstract

Let R be a commutative ring with unity and M be an R-module. In this study, we construct the \tilde{Spec}(M) topology using the prime spectrum of module M and multiplicatively closed subsets of R with the closed sets \tilde{V}(S)={P \in Spec(M) : (P : M) \cap S_i \neq \emptyset for all i \in I} with the open sets \tilde{D}(S_i):={P \in Spec(M) : (P : M) \cap S_i = \emptyset} where S = {S_i}_{i \in I} is a family of multiplicatively closed subsets of R. We investigate connections between the algebraic properties of R-module M and the topological properties of \tilde{Spec}(M). We examine specifically the separation axioms, connectivity, nested and Lindelöf property together with quasi-compactness as well as the isolated, closure, interior and limit points of tilde{Spec}(M). Moreover, in the last section, we provide an example of a Lindelöf space which is not quasi-compact by means of \tilde{Spec}(M).

Theorems & Definitions (52)

• Theorem 1
• proof
• Theorem 2
• proof
• Proposition 1
• proof
• Example 1
• Lemma 1
• proof
• Lemma 2