On Divisor Topology of Commutative Rings

Let R\be an integral domain and $R^{\mathrm{\#}}$ the set of all nonzero nonunits of R.\For every elements $a,b\in R^{\mathrm{\#}},$ we define $a\sim b$ if and only if $aR=bR,$ that is, $a$ and $b$ are associated elements. Suppose that $EC(R^{\mathrm{\#}})$ is the set of all equivalence classes of R^{\#}\according to $\sim$.\Let U_{a}=\{[b]\in EC(R^{\#}):b\divides $a\}$ for every $a\in R^{\mathrm{\#}}\mathrm{.}$ Then we prove that the family $\{U_a{\}}_{a\in R^{\mathrm{\#}}}$ 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 $D(R)\mathrm{.}$ 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 $D(R)$.