A Structure Theorem for Noetherian P.I. Rings with Global Dimension Two

LetAbe the Artin radical of a Noetherian ringRof global dimension two. We show thatA = ReRwhereeis an idempotent;Acontains a heredity chain of ideals and the global dimensions of the ringsR/AandeRecannot exceed two. Assume further thanRis a polynomial identity ring. LetPbe a minimal prime ideal ofR. ThenP = P2and the global dimension ofR/Pis also bounded by two. In particular, if the Krull dimension ofR/Pequals two for all minimal primesPthenRis a semiprime ring. In general, every clique of prime ideals inRis finite and in the affine caseRis a finite module over a commutative affine subring. Additionally, whenA = 0, the ringRhas an Artinian quotient ring and we provide a structure theorem which shows thatRis obtained by a certain subidealizing process carried out on rings involving Dedekind prime rings and other homologically homogeneous rings.