LetRbe a ring and ρ a right ideal ofRwith zero right annihilate. Then ρ andRhave the same left Utumi quotient ring. We study the lifting properties of GPIs and some chain conditions inherited by such right ideals. Next, we prove a generalization of Chatters’ theorem. Precisely, we show that ifRis a right nonsingular ring with finite right Goldie dimension and possesses a right ideal ρ such that both ρ andlR(ρ) are PI-rings, then the right Utumi quotient ring ofRis also a Pi-ring.