A structure theorem for Noetherian P. I. rings with global dimension two
The main purpose of this paper is to give a structure theorem for Noetherian rings of global dimension two which satisfy a polynomial identity (PI). (The Noetherian PI rings of global dimension 1 are well understood, being a direct sum of an Artinian hereditary ring and finitely many prime hereditary classical orders over Dedekind domains.) Let \(R\) be a Noetherian PI ring of global dimension 2. Then the Artin radical \(A(R)\) is generated by an idempotent of \(R\), and \(R/A(R)\) has global dimension at most 2. Thus one analyses separately the structure of \(A(R)\), and of a ring \(R\) as above, but with \(A(R)=0\). For the former, there is a set of results reminiscent of, and developed from, the theory of quasihereditary rings, specifically [\textit{V. Dlab} and \textit{C. M. Ringel}, Ill. J. Math. 33, No. 2, 280-291 (1989; Zbl 0666.16014)]. In particular, \(A(R)\) is a right and left projective module, and contains a hereditary chain of ideals. Then, a ring \(R\) which (*) is Noetherian PI with \(A(R)=0\) and global dimension 2 is shown to have an Artinian quotient ring and to be obtained by finitely many iterations of an idealizer process from a finite direct sum of \(d\) rings \(R_i\) which satisfy (*) and are of explicitly described type. Conversely, this process can be reversed, to get \(\bigoplus^d_{i=1}R_i\) from \(R\). Along the way towards this result many interesting subsidiary results are proved, corollaries deduced, and examples provided.
Noetherian PI rings of global dimension 2
scientific article; zbMATH DE number 1295458
scientific article