Orders in primitive rings with non-zero socle and Posner's theorem

Combined with a theorem of Xaplansky, Posner's theorem can be reformulated as follows: A ring is a prime PI-ring if and only if it is an order in a primitive PI-ring. In this paper we prove a similar characterization for prime GPI-rings: A ring is a prime GPI-ring ii and only ii it is an ''order'' in a primitive GPI-ring. In order that this statement be valid, we have to modify, however, the classical notion of ring oi quotients, Our notion is based on that of Fountain and Gould [6] but is weaker than theirs, This new kind of quotient rings makes it possible to extend also Goldie's theorem to non-singular prime rings with uniform one-sided ideals (see Theorem 1 below), and this fact has a key role in proving Posner's theorem for the GPI case.