Global dimension in Noetherian rings and rings with Gabriel and Krull dimension (Q1199349)

This paper contains some very nice formulas for computing the left global dimension, denoted by \(\text{lgl}\dim(R)\), of a ring \(R\) for a ring with Gabriel dimension (say, \(\text{G-}\dim R=\beta)\) in terms of the projective or injective dimensions of certain cyclic left \(R\)-modules. For an ordinal \(\alpha\), a nonzero \(R\)-module \(C\) is \(\alpha\)-simple if \(\text{G-}\dim C=\alpha\), but \(\text{G-}\dim C/N<\alpha\) for every \(0\neq N\subseteq C\). Then \(\text{lgl}\dim(R)=\sup\{\text{pd }C\mid C\) is an \(\alpha\)-simple cyclic \(R\)-module, \(\alpha<\beta\}\). If \(R\) is left Noetherian, then \(\text{lgl}\dim(R)=\sup\{\text{id}(C)\mid C\) is an \(\alpha\)-simple cyclic \(R\)-module with \(\text{id}(C)=\text{id}(C')\) for all \(0\neq C'\subseteq C\), \(\alpha<\beta\}\). Examples are given to illustrate the usefulness of these results.