Homological properties of generic matrix rings
Although generic matrix rings play a very important role in the theory of rings with polynomial identity, little is known about their homological properties. It is suspected that their global dimension is always infinite, and the authors establish this for the ring \(R=k_{(2)}\{a,b\}\) of two generic \(2\times 2\) matrices over an infinite field k. The trace ring T of R is shown to be an iterated Ore extension with \(gl.\dim (T)=5\), to be a maximal order, and to be stably free. Furthermore, if \(A=R\) or T, then there exists a non-free projective right ideal P satisfying \(P\oplus A\simeq A\oplus A\). It is announced that R is also stably free, a proof will be furnished by Severino Collier Coutinho in a forthcoming paper.
scientific article; zbMATH DE number 3929206
scientific article