Global dimension of rings with krull dimension

Abstract

The left global dimension of a semiprime ring, with left Krull dimension α≥1 is found to be the supremum of the projective dimensions of the p-critical cyclic modules where β≤α. A similar result is true for upper triangular matrix rings whose entries come from a domain with Krull dimension. In addition if R is a a ring of the form where S is a semiprime ring with left Krull dimension α≥1, T is any ring with l.K dim T≤α, and A is an S-T bimodule such that sA has Krull dimension then the left global dimension of R is the supremum of the projective dimensions of the -critical cyclic left R-modules where β<αa. These results are used to compute homological dimensions of rings with Krull dimension. Some analogues are given for weak dimension and for rings with Gabriel dimension.