Abstract.-This paper considers a generalization of Artin's conjecture on primes with prescribed primitive roots. The main result provides & necessary and sufficient condition for the conjectural density of certain sets of primes to be non-zero. As an application a theorem about the existence of a euclidean algorithm in rings of arithtnetic type is proved.Key_word£: Artin's conjecture, primitive roots, Euclid's algorithm.12A75, 10HI5, 12A45, 13F10.