Projective Prime Ideals and Localisation in PI-Rings

The results here generalise 2, Proposition 4.3] and 9, Theorem5.11]. We shall prove the following. THEOREM A. Let R be a Noetherian PI-ring. Let P be a non-idempotentprime ideal of R such that P_R is projective. Then P is leftlocalisable and R_P is a prime principal left and right idealring. We also have the following theorem. THEOREM B. Let R be a Noetherian PI-ring. Let M be a non-idempotentmaximal ideal of R such that M_R is projective. Then M has theleft AR-property and M contains a right regular element of R.