A NOTE ON DENSITY AND THE DIRICHLET CONDITION

Abstract

Motivated by topological approaches to Euclid and Dirichlet's theorems on infinitude of primes, we introduce and study [Formula: see text]-coprime topologies on a commutative ring R with an identity and without zero divisors. For infinite semiprimitive commutative domain R of finite character (i.e. every nonzero element of R is contained in at most finitely many maximal ideals of R), we characterize its subsets A for which the Dirichlet condition, requiring the existence of infinitely many pairwise nonassociated elements from A in every open set in the invertible topology, is satisfied.