A note on utumi quotient rings

Let R be a ring and ρ a right ideal of R with zero right annihilate. Then ρ and R have 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 if R is a right nonsingular ring with finite right Goldie dimension and possesses a right ideal ρ such that both ρ and l _R(ρ) are PI-rings, then the right Utumi quotient ring of R is also a Pi-ring.