Let A be the Artin radical of a Noetherian ring R of global dimension two. We show that A = ReR where e is an idempotent; A contains a heredity chain of ideals and the global dimensions of the rings R/A and eRe cannot exceed two. Assume further than R is a polynomial identity ring. Let P be a minimal prime ideal of R. Then P = P2 and the global dimension of R/P is also bounded by two. In particular, if the Krull dimension of R/P equals two for all minimal primes P then R is a semiprime ring. In general, every clique of prime ideals in R is finite and in the affine case R is a finite module over a commutative affine subring. Additionally, when A = 0, the ring R has an Artinian quotient ring and we provide a structure theorem which shows that R is obtained by a certain subidealizing process carried out on rings involving Dedekind prime rings and other homologically homogeneous rings.
Bibliographical noteFunding Information:
* This work was partially supported by an EPSRC grant at the University of Warwick. The first author thanks the Mathematics Institute at the University of Warwick for its hospitality.
ASJC Scopus subject areas
- Algebra and Number Theory