Simón Pinero, Juan Jacobo Global dimension in Noetherian rings and rings with Gabriel and Krull dimension. (English) Zbl 0773.16003 Publ. Mat., Barc. 36, No. 1, 189-195 (1992). 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. Reviewer: M.L.Teply (Milwaukee) Cited in 2 Reviews MSC: 16E10 Homological dimension in associative algebras 16P60 Chain conditions on annihilators and summands: Goldie-type conditions 16P40 Noetherian rings and modules (associative rings and algebras) Keywords:homological dimension; torsion theory; cocritical module; left global dimension; Gabriel dimension; projective or injective dimensions; cyclic left \(R\)-modules; left Noetherian PDFBibTeX XMLCite \textit{J. J. Simón Pinero}, Publ. Mat., Barc. 36, No. 1, 189--195 (1992; Zbl 0773.16003) Full Text: DOI EuDML