Extension closure of k-torsionfree modules

Abstract

Let Λ be a left and right Noetherian ring. For a positive integer k, we give an equivalent condition that flat dimensions of the first k terms in the minimal injective resolution of Λ are less than or equal to k. In this case we show that the subcategory consisting of k-torsionfree modules is extension closed. Moreover we prove that for a Noetherian algebra every subcategory consisting of i-torsionfree modules is extension closed for any 1 ≤ i ≤ k if and only if every subcategory consisting of i-th syzygy modules is extension closed for any 1 ≤ i ≤ k. Our results generalize the main results in Auslander and Reiten [4].

Keywords:

• extension closed
• k-torsionfree modules
• Noetherian rings
• flat dimension

1991 Mathematics Subject Classification:

• 16E10
• 16D50
• 16P40

¹The author’s research was supported by the National Science Foundation of People’s Republic of Chins and Japanese Administration of Education as Scholarship in Yamanashi University.

¹The author’s research was supported by the National Science Foundation of People’s Republic of Chins and Japanese Administration of Education as Scholarship in Yamanashi University.

Notes

¹The author’s research was supported by the National Science Foundation of People’s Republic of Chins and Japanese Administration of Education as Scholarship in Yamanashi University.