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Global dimension in Noetherian rings and rings with Gabriel and Krull dimension. (English) Zbl 0773.16003

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.

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)
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