On Divisor Topology of Commutative Rings

Let $R\$be an integral domain and $R^{\#}$ the set of all nonzero nonunits of $R.\$For every elements $a,b\in R^{\#},$ we define $a\sim b$ if and only if $aR=bR,$ that is, $a$ and $b$ are associated elements. Suppose that $EC(R^{\#})$ is the set of all equivalence classes of $R^{\#}\$according to $\sim$.$\$Let $U_{a}=\{[b]\in EC(R^{\#}):b\$divides $a\}$ for every $a\in R^{\#}.$ Then we prove that the family $\{U_{a}\}_{a\in R^{\#}}$ becomes a basis for a topology on $EC(R^{\#}).\$This topology is called divisor topology of $R\$and denoted by $D(R).\$We investigate the connections between the algebraic properties of $R\$and the topological properties of$\ D(R)$. In particular, we investigate the seperation axioms on $D(R)$, first and second countability axioms, connectivity and compactness on $D(R)$. We prove that for atomic domains $R,\$the divisor topology $D(R)\$is a Baire space. Also, we characterize valution domains $R$ in terms of nested property of $D(R).$ In the last section, we introduce a new topological proof of the infinitude of prime elements in a UFD and integers by using the topology $D(R)$.