Quasi Divisor Topology of Modules over Domains

Abstract

Let $E$ be a module over a domain $A$, and $W(E{)}^{\#}=W(E)-ann(E)$ where $W(E)=\{a\in A:aE\ne E\}$. We define an equivalence relation $\sim$ on $W(E{)}^{\#}$ as follows: $a\sim b$ if and only if $aE=bE$ for any $a,b\in W(E{)}^{\#}$ and denote $EC(W(E{)}^{\#})$ to be the set of all equivalence classes $[a]$ of $W(E{)}^{\#}$. We first show that the family $\{U_a{\}}_{a\in W(E{)}^{\#}}$ generates a topology which we called the quasi divisor topology of $A$-module $E$ denoted by $q{D}_A(E)$ where $U_a=\{[b]\in EC(W(E{)}^{\#}):aE\subseteq bE\}$ for every $a\in W(E{)}^{\#}$. This paper examines the connections between topological properties of the quasi divisor topology $q{D}_A(E)$ and algebraic properties of $A$-module $E$. These include each separation axioms, compactness, connectedness and first and second countability. Also, we characterize some important class of rings/modules such as divisible modules and uniserial modules by means of $q{D}_A(E)$. Furthermore, we introduce quasi second modules and study its algebraic properties to decide when $q{D}_A(E)$ is a $T_1$-space.