Abstract:

Let R be a ring, n a fixed nonnegative integer and $\mathscr{G} \mathscr{I}_n$ (resp., $\mathscr{G} \mathscr{F}_n$) the class of all left (resp., right) R-modules of Gorenstein AC injective (resp., Gorenstein AC flat) dimension at most n. A left R-module M (resp., right R-module M) is called n-copure Gorenstein AC injective (resp., n-copure Gorenstein AC flat) if Ext_R¹ (N, M)=0 (resp., Tor₁^R (M, N)=0) for any N ∈ $\mathscr{G} \mathscr{I}_n$. It is proven that a finitely presented right R-module M is n-copure Gorenstein AC flat if and only if M is a cokernel of a $\mathscr{G} \mathscr{F}_n$- preenvelope K→F of a right R-module K with F flat.

Key words: n-copure Gorenstein AC-injective module, n-copure Gorenstein AC-flat module, $\mathscr{G} \mathscr{F}_n$-preenvelope