Is a cockroach considered najas (impure), and what is the ruling concerning things it walks over?

Cockroaches that originate from pure (tahir) sources are pure, whether alive or dead. Those that originate from impure (najis) sources are impure, whether alive or dead. The passage of cockroaches over the ground or carpets does not render them impure due to the difficulty of avoiding them, unless they leave a trace of impurity, in which case it must be purified.