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

✍ Scribed by Amiram Braun; Charudatta R Hajarnavis

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 336 KB
• Volume: 215
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1998.7737

✦ Synopsis

Let A be the Artin radical of a Noetherian ring R of global dimension two. We show that A s ReR where e is an idempotent; A contains a heredity chain of ideals and the global dimensions of the rings RrA 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 s P 2 and the global dimension of RrP is also bounded by two. In particular, if the Krull dimension of RrP 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 s 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.