On n-FI-Injective and n-FI-Flat Modules

Abstract

Let R be a ring, n a fixed non-negative integer and ℱℐ _n (ℱ _n ) the class of all left (right) R-modules of FP-injective (flat) dimension at most n. A left R-module M (resp., right R-module N) is called n-FI-injective (resp., n-FI-flat) if (resp., ) for any F ∈ ℱℐ _n . It is proved that a left R-module M is n-FI-injective if and only if M is a kernel of an ℱℐ _n -precover f: A → B of a left R-module B with A injective. For a left coherent ring R, it is shown that a finitely presented right R-module M is n-FI-flat if and only if M is a cokernel of an ℱ _n -preenvelope K → F of a right R-module K with F flat. Some known results are extended. Finally, we investigate n-FI-injective and n-FI-flat modules over left coherent rings with FP-id( _R R) ≤ n.

Key Words:

• Coherent ring
• FP-injective dimension
• n-FI-injective module
• n-FI-flat module
• Precover
• Preenvelope

2010 Mathematics Subject Classification:

• 16E10
• 16E30
• 16D50

ACKNOWLEDGMENTS

I would like to express my gratitude to Professor Fanggui Wang for his support, and constant encouragement. Furthermore, I would like to thank Professor Ivan Shestakov and the referee for their useful comments on this paper. This work was partially supported by NSFC (No. 11171240), and the Scientific Research Foundation of CUIT (No. KYTZ201201).

Notes

Communicated by I. Shestakov.