The results here generalise [2, Proposition 4.3] and [9, Theorem 5.11]. We shall prove the following.\ud \ud THEOREM A. Let R be a Noetherian PI-ring. Let P be a non-idempotent prime ideal of R such that PR is projective. Then P is left localisable and RP is a prime principal left and right ideal ring.\ud \ud We also have the following theorem.\ud \ud THEOREM B. Let R be a Noetherian PI-ring. Let M be a non-idempotent maximal ideal of R such that MR is projective. Then M has the left AR-property and M contains a right regular element of R
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