Gorenstein Homological Dimension with Respect to a Semidualizing Module and a Generalization of a Theorem of Bass

Abstract

Let C be a semidualizing module for a commutative ring R. In this paper, we study the resulting modules of finite G _C -projective dimension in Bass class, showing that they admit G _C -projective precover. Over local ring, we prove that dim _R (M) ≤ 𝒢ℐ _C  − id _R (M) for any nonzero finitely generated R-module M, which generalizes a result due to Bass.

Key Words:

• Auslander classes
• Bass classes
• G _C -projective
• G _C -injective
• Precover
• Preenvelope
• Semidualizing

2010 Mathematics Subject Classification:

• 13D05
• 13D07
• 18G20

ACKNOWLEDGMENTS

The authors are very grateful to the referee for several suggestions and comments that greatly improved the paper.

The research of S. Yassemi was in part supported by a grant from IPM No. 92130214.

Notes

Communicated by T. Albu.