n-Coherent Rings

Gorenstein $$n$$ n -flat dimension and Tate homology

“…right) R-module M is called n-flat [9] if T or R 1 (N , M) = 0 holds for all finitely presented right (resp. left) R-modules N with projective dimension ≤ n and a left (resp.…”

“…left) R-modules N with projective dimension ≤ n and a left (resp. right) R-module M is called n-absolutely pure [9] if E xt 1 R (N , M) = 0 holds for all finitely presented left (resp. right) R-modules N with projective dimension ≤ n. Also recall that a ring R is called right n-coherent [9] (for integers n > 0 or n = ∞) if every finitely generated submodule of a free right R-module whose projective dimension is ≤ n − 1 is finitely presented.…”

“…right) R-module M is called n-absolutely pure [9] if E xt 1 R (N , M) = 0 holds for all finitely presented left (resp. right) R-modules N with projective dimension ≤ n. Also recall that a ring R is called right n-coherent [9] (for integers n > 0 or n = ∞) if every finitely generated submodule of a free right R-module whose projective dimension is ≤ n − 1 is finitely presented. A left R-module U is said to be n-cotorsion [12] if E xt 1 R (F, U ) = 0 for all n-flat left R-modules F, i.e., U ∈ F ⊥ n , where F n is the class of all n-flat left R-modules.…”

n-Prüfer domains

“…Since E was arbitary, Torf (TV, .4) = 0, which implies that A is n-flat. D In Lee [4], a domain R was called n-coherent if every finitely generated torsion-free .R-module of projective dimension < n -1 is finitely presented. n-Priifer domains are trivially n-coherent, and we are going to show that the converse holds when 1-flat modules are n-flat.…”

“…PROOF: In view of Lemma 6, we have only to prove (b) implies (a). Let D be a 1-absolutely pure R-module and N a finitely presented it-module of projective dimension < n. Since R is n-coherent, N has a finite projective resolution (see Lee [4]). In the natural isomorphism…”