Gorenstein Test Modules^#

^#Communicated by I. Swanson.

Let (R, 𝔪) be a commutative Noetherian local ring. We exhibit certain modules T over R which test G-dimension of a finitely generated R-module M with finite G-dimension in the following sense: if for all j ≥ i, where i is a positive integer, then G-dim _R  M < i. Modules with the property like T will be called Gorenstein test modules (G-test modules for short). It is known that R itself is a G-test module. We show that k, the residue field of R, also tests G-dimension. Some more examples of G-test modules are introduced.

Key Words:

• Gorenstein dimension
• Gorenstein injective modules
• Gorenstein ring

2000 Mathematics Subject Classification:

• 13D05
• 13D45
• 13H10
• 13D03
• 55N35

ACKNOWLEDGMENTS

The authors are deeply grateful to the referee for his/her useful pointed comments on the article. This research was in part supported by a grant from IPM (No. 82130113 and No. 82130118).

Notes

^#Communicated by I. Swanson.