Research Information System
Ricerche di Matematica, vol.74, no.4, pp.2417-2429, 2025 (SCI-Expanded, Scopus)
Let R be an integral domain and R# the set of all nonzero nonunits of R. For every element a,b∈R#, we define a∼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 ∼. Let Ua={[b]∈EC(R#):b divides a} for every a∈R#. Then we prove that the family {Ua}a∈R# becomes a basis for a topology on EC(R#). This topology is called the divisor topology of R and is denoted by D(R). We investigate the connections between the algebraic properties of R and the topological properties ofD(R). In particular, we investigate the separation 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 valuation domains R in terms of the 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).