ORDERS IN PRIMITIVE RINGS WITH NONZERO SOCLE AND POSNERS THEOREM

Combined with a theorem of Xaplansky, Posner's theorem can be reformul ated as follows: A ring is a prime PI-ring if and only if it is an ord er in a primitive PI-ring. In this paper we prove a similar characteri zation 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 stateme nt 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] bu t is weaker than theirs, This new kind of quotient rings makes it poss ible 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 ke y role in proving Posner's theorem for the GPI case.