Abstract
In the last years (Gorenstein) homological dimensions relative to a semidualizing module C have been subject of several works as interesting extensions of (Gorenstein) homological dimensions. In this paper, we extend to the noncommutative case the concepts of G C -projective module and dimension, weakening the condition of C being semidualizing as well. We prove that indeed they share the principal properties of the classical ones and relate this new dimension with the classical Gorenstein projective dimension of a module. The dual concepts of G C -injective modules and dimension are also treated. Finally, we show some interesting interactions between the class of G C -projective modules and the Bass class associated to C on one side, and the class of G\({_{C^{\vee}}}\) -injective modules (C ∨ = Hom R (C, E) where E is an injective cogenerator in R-Mod) and the Auslander class associated to C in the other.
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This work started when D. Bennis was visiting the University of Almería. D. Bennis would like to express his sincere thank to the warm hospitality and the excellent working conditions. This research was partially supported by the grant MTM2011-27090 from Ministerio de Ciencia e Innovación of Spain.
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Bennis, D., García Rozas, J.R. & Oyonarte, L. Relative Gorenstein Dimensions. Mediterr. J. Math. 13, 65–91 (2016). https://doi.org/10.1007/s00009-014-0489-8
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DOI: https://doi.org/10.1007/s00009-014-0489-8