Zariski Topologies for Coprime and Second Submodules

Let $M$ be a non-zero module over an associative (not necessarily commutative) ring. In this paper, we investigate the so-called \emph{second} and \emph{coprime} submodules of $M.$ Moreover, we topologize the spectrum $% \mathrm{Spec}^{\mathrm{s}}(M)$ of second submodules of $M$ and the spectrum $% \mathrm{Spec}^{\mathrm{c}}(M)$ of coprime submodules of $M,$ study several properties of these spaces and investigate their interplay with the algebraic properties of $M.$

💡 Research Summary

The paper investigates two novel classes of submodules—second and coprime—within a non‑zero module (M) over an arbitrary (not necessarily commutative) ring (R). A second submodule (N\subseteq M) is defined by the condition that for any (r\in R), (rN=0) forces (r) to belong to (\operatorname{Ann}_R(M)). In other words, (N) is “second” in the sense that its annihilator is already the annihilator of the whole module. A coprime submodule (C\subseteq M) (different from (M) itself) satisfies: if (rM\subseteq C) then (r\in\operatorname{Ann}_R(M/C)). This mirrors the classical notion of a coprime ideal, but lifted to the module level.

From these definitions the authors construct two spectra: \