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