GRADE AND GORENSTEIN DIMENSION

Abstract

Let R be a commutative Noetherian ring and let M be a finite (that is, finitely generated) R-module. The notion grade of M, grade M, has been introduced by Rees as the least integer t ≥ 0 such that Ext ^t _R (M,R) ≠ 0, see [11]. The Gorenstein dimension of M, G-dim M, has been introduced by Auslander as the largest integer t ≥ 0 such that Ext ^t _R (M, R) ≠ 0, see [3]. In this paper the R-module M is called G-perfect if grade M = G-dim M. It is a generalization of perfect module. We prove several results for the new concept similar to the classical results.

Keywords:

• 1991 Mathematics Subject Classification: 13H10, 13C15
• Perfect module
• Grade of module
• Cohen–Macaulay module
• Gorenstein ring

ACKNOWLEDGMENTS

The authors would like to thank the University of Tehran for the financial support and the Institute for studies in Theoretical Physics and Mathematics (IPM) for the facilities offered during the preparation of this paper. The authors would also like to thank the referee for his useful comments.