Bennis, Driss; Mahdou, Najib Global Gorenstein dimensions of polynomial rings and of direct products of rings. (English) Zbl 1186.13007 Houston J. Math. 35, No. 4, 1019-1028 (2009). Over any ring \(R\), E. E. Enochs and O. M. G. Jenda [Math. Z. 220, No. 4, 611–633 (1995; Zbl 0845.16005)] introduced the class of Gorenstein projective (left) \(R\)-modules. The Gorenstein projective dimension \(\text{Gpd}_RM\) of a (left) \(R\)-module \(M\) is the length of the shortest resolution of \(M\) consisting of Gorenstein projective (left) \(R\)-modules. The (left) Gorenstein global dimension of \(R\), denoted by \(\text{G-gldim}(R)\), is the supremum of \(\text{Gpd}_RM\), where \(M\) ranges over all (left) \(R\)-modules.In this paper it is proved that \(\text{G-gldim}(R[X]) = \text{G-gldim}(R)+1\) if \(R\) is commutative. It is also proved that \(\text{G-gldim}(R\times S) = \max\{\text{G-gldim}(R),\text{G-gldim}(S)\}\) if \(R\) and \(S\) are commutative.Similar results are obtained for the weak Gorenstein global dimension. Reviewer: Henrik Holm (Copenhagen) Cited in 6 Documents MSC: 13D02 Syzygies, resolutions, complexes and commutative rings 13D05 Homological dimension and commutative rings 13D07 Homological functors on modules of commutative rings (Tor, Ext, etc.) Keywords:Gorenstein global dimension; weak Gorenstein global dimension Citations:Zbl 0845.16005 × Cite Format Result Cite Review PDF Full Text: arXiv Link