Coherent rings and absolutely pure precovers

On uniformly $S$-absolutely pure modules

“…(3) there exists an element s ∈ S satisfying that for any given commutative diagram with F finitely generated free and K a finitely generated submodule of F , there exists a homomorphism η : F → A such that sα = ηi; 4) there exists an element s ∈ S satisfying that for any finitely presented Rmodule N, the induced sequence 0 → Hom R (N, A) → Hom R (N, B) → Hom R (N, C) → 0 is S-exact with respect to s. (1). So there is an element s ∈ S such that sKer(1…”

“…A ring R is von-Neumann regular if and only if any R-module is absolutely pure ([11,Theorem 5]). A ring R is coherent if and only if the class of absolutely pure R-modules is closed under direct limits, if and only if the class of absolutely pure R-modules is a (pre)cover ( [14,Theorem 3.2], [4,Corollary 3.5]).…”

Some remarks on Lucas modules

“…Let f : C → M be a C-cover of M. If for any f ′ : C ′ → M with C ′ ∈ C, there exists a unique g : C ′ → C such that f ′ = f • g. Then f is said to have the unique mapping property. Let L be the class of all Lucas modules over a ring R. An L-(pre)cover of an R-module M is said to be a Lucas (pre)cover of M. The author et al [14] put [2,Theorem 3.4], [7,Theorem 3.4.8] and [6, Theorem 2.5] together in the following lemma.…”