On Divisor Topology of Modules over Domains

Let M\be a module over a domain $R$ and $M^{\mathrm{\#}}=\{0\mathrm{\ne }m\in M:Rm\mathrm{\ne }M\}$ be the set of all nonzero nongenerators of M.\Consider following equivalence relation $\sim$ on $M^{\mathrm{\#}}$ as follows: for every $m,n\in M^{\mathrm{\#}},m\sim n$ if and only if Rm=Rn.\Let $EC(M^{\mathrm{\#}})$ be the set of all equivalence classes of $M^{\mathrm{\#}}$ with respect to $\sim$. In this paper, we construct a topology on $EC(M^{\mathrm{\#}})$ which is called divisor topology of M\and denoted by $D(M)\mathrm{.}$ Actually, $D(M)$ is extension of the divisor topology $D(R)$ over domains in the sense of Yiğit and Koc to modules. We investigate separation axioms $T_i$ for every $0\le i\le 5,$ first and second countability, connectivity, compactness, nested property, and Noetherian property on $D(M)$. Also, we characterize some important classes of modules such as uniserial modules, simple modules, vector spaces, and finitely cogenerated modules in terms of $D(M)$. Furthermore, we prove that $D(M)$ is a Baire space for factorial modules. Finally, we introduce and study pseudo simple modules which is a new generalization of simple modules, and use them to determine when $D(M)$ is a discrete space.