A Topology on S-Spectrum of a Module

Yıldız E. , Ersoy B. A. , Tekir Ü.


  • Publication Type: Conference Paper / Unpublished
  • City: İstanbul
  • Country: Turkey


Prime ideals/submodules and their generalizations play a significant role in Commutative Algebra and Algebraic Geometry. These structures help to characterize some rings and modules and also they have many applications in some branches of mathematics like Topology and Graph Theory. Let 𝑅 be a commutative ring with identity and 𝑆 be a multiplicatively subset of 𝑅, 𝑀 be an 𝑅- module and 𝑃 be a proper submodule of 𝑀 with (𝑃: 𝑀) ∩ 𝑆 = ∅. Then 𝑃 is said to be an Sprime submodule if there exists a fixed 𝑠 ∈ 𝑆 such that 𝑎𝑚 ∈ 𝑃 for some 𝑎 ∈ 𝑅, 𝑚 ∈ 𝑀 implies that either 𝑠𝑎 ∈ (𝑃: 𝑀) or 𝑠𝑚 ∈ 𝑃. The set of all S-prime submodules of 𝑀 is denoted by 𝑆𝑝𝑒𝑐𝑆 (𝑀). We define the sets 𝑉𝑠(𝐾) = {𝑃 ∈ 𝑆𝑝𝑒𝑐𝑆(𝑀): 𝑠(𝐾: 𝑀) ⊆ (𝑃: 𝑀), ∃𝑠 ∈ 𝑆} and 𝑉𝑠 ∗ (𝐾) = {𝑃 ∈ 𝑆𝑝𝑒𝑐𝑆(𝑀): 𝑠𝐾 ⊆ 𝑃, ∃𝑠 ∈ 𝑆}. The former set satisfies all the axioms to be a closed set in a topological space. Then we construct a topology by using this type of closed sets and the topology which is called S-Zariski topology and this topology is a generalization of classical Zariski topology. The latter set may not lead to a topology; however, in some special structures we can construct a topology by using this set and this topology is called quasi S-Zariski topology. In this study, we investigate many properties of S-Zariski topology on a module and introduce the topology of some special modules such as multiplication modules. Moreover, we construct some continuous maps and give the relations between these maps.