Prime matrix rings

where the ei1 are the usual unit matrices. For example, we could select n left ideals Al, * * *, An of either F or a subring of F and then let Fij=Aj, i, j=1, . I n. If F is a (skew) field and the Fij satisfying (1) are all nonzero, then R defined by (2) is easily shown to be a prime ring. The main result of this paper (1.3) is that if F is a right ring of quotients of Fil then (F)X is a right ring of quotients of R and there exists a subring K of F and a nonzero diagonal matrix dCR such that (K)n is a subring of dRd-1 and F is a ring of quotients of K. This result is used to give new proofs of the Faith-Utumi theorem [2] and of Goldie's theorem [l].