ALMOST F-INJECTIVE MODULES AND ALMOST FLAT MODULES

A left R-module M is called almost F-injective, if every R-homomorphism from a finitely presented left ideal to M extends to a homomorphism of R to M. A right R-module V is said to be almost flat, if for every finitely presented left ideal I, the canonical map V ⊗ I → V ⊗R is monic. A ring R is called left almost semihereditary, if every finitely presented left ideal of R is projective. A ring R is said to be left almost regular, if every finitely presented left ideal of R is a direct summand of RR. We observe some characterizations and properties of almost F-injective modules and almost flat modules. Using the concepts of almost F-injectivity and almost flatness of modules, we present some characterizations of left coherent rings, left almost semihereditary rings, and left almost regular rings. Mathematics Subject Classification (2010): 16E50, 16E60, 16D40, 16D50, 16P70


Introduction
Throughout this paper, R denotes an associative ring with identity and all modules considered are unitary.For any R-module M, M + = Hom(M, Q/Z) will be the character module of M.
Recall that a left R-module A is said to be finitely presented if there is an exact sequence F 1 → F 0 → A → 0 in which F 1 , F 0 are finitely generated free left R-modules, or equivalently, if there is an exact sequence P 1 → P 0 → A → 0, where P 1 , P 0 are finitely generated projective left R-modules.Let n be a positive integer.Then a left R-module M is called n-presented [2] if there is an exact sequence of left R-modules which every F i is a finitely generated free, equivalently, projective left R-module.A left R-module M is said to be FP-injective [9] if Ext 1 (A, M) = 0 for every finitely presented left R-module A. FP-injective modules are also called absolutely pure modules [7].A left R-module M is called F-injective [4] if every R-homomorphism from a finitely generated left ideal to M extends to a homomorphism of R to M, or equivalently, if Ext 1 (R/I, M) = 0 for every finitely generated left ideal I.A right R-module M is flat if and only if Tor 1 (M, A) = 0 for every finitely presented left R-module A if and only if Tor 1 (M, R/I) = 0 for every finitely generated left ideal I.We recall also that a ring R is called left coherent if every finitely generated left ideal of R is finitely presented.Coherent rings and their generations have been studied by many authors (see, for example, [1,2,6,7,9,10]).
In this paper, we shall extend the concepts of F-injective modules and flat modules to

Almost F-injective modules and almost flat modules
We first extend the concept of F-injective modules as follows.
Recall that a submodule A of a left R-module B is said to be a pure submodule if for every right R-module M, the induced map M ⊗ R A → M ⊗ R B is monic, or equivalently, every finitely presented left R-module is projective with respect to the exact sequence 0 → A → B → B/A → 0. In this case, the exact sequence 0 → A → B → B/A → 0 is called pure.It is well known that a left R-module M is FP-injective if and only if it is pure in every module containing it as a submodule.We call a short exact sequence of left R-modules 0 → A → B → C → 0 almost pure if for every finitely presented left ideal I, R/I is projective with respect to this sequence.In this case, we call A an almost pure submodule of B.
Next, we give some characterizations of almost F-injective modules.
Theorem 2.2.Let M be a left R-module.Then the following statements are equivalent.
(3) M is injective with respect to every exact sequence 0 → C → B → R/I → 0 of left R-modules with I a finitely presented left ideal.
(4) M is injective with respect to every exact sequence 0 → K → P → R/I → 0 of left R-modules with P projective and I a finitely presented left ideal.
(5) Every exact sequence of left R-modules 0 → M → M → M → 0 is almost pure.
Clearly, a right R-module V is almost flat if and only if Tor 1 (V, R/I) = 0 for every finitely presented left ideal I. Proposition 2.6.Let {V i | i ∈ I} be a family of right R-modules.Then the following statements are equivalent.
Theorem 2.7.The following are true for any ring R.
(3) The class of almost F-injective left R-modules is closed under pure submodules, pure quotients, direct sums, direct summands, direct products and direct limits.(2) This follows from the duality formula where M is a right R-module and N is a left R-module.By the isomorphism formula we see that the class of almost flat right R-modules is closed under direct limits.
Theorem 2.8.The following statements are equivalent for a ring R.
(3) Every almost F-injective left R-module is FP-injective.Let F be a class of left (right) R-modules and M a left (right) R-module.Following [3], we say that a homomorphism ϕ : M → F where F ∈ F is an F -preenvelope of M if for any morphism f : M → F with F ∈ F , there is a g : F → F such that gϕ = f .An F -preenvelope ϕ : M → F is said to be an F -envelope if every endomorphism g : F → F such that gϕ = ϕ is an isomorphism.Dually, we have the definitions of F -precovers and F -covers.F -envelopes (F -covers) may not exist in general, but if they exist, they are unique up to isomorphism.
Theorem 2.9.The following hold for any ring R.
(1) Every left R-module has an almost F-injective cover and an almost F-injective preenvelope.
(2) Every right R-module has an almost flat cover and an almost flat preenvelope.(2) is similar to that of (1).
( Consider the following commutative diagram: Since τ 1 and τ 2 are isomorphisms, Hom(M, B + ) → Hom(M, A + ) is an epimorphism.So The other is similar.
Proposition 2.10.The following statements are equivalent for a ring R.
(2) Every left R-module has an epic almost F-injective cover.
(3) Every right R-module has a monic almost flat preenvelope.
(4) Every injective right R-module is almost flat.
(5) Every FP-injective right R-module is almost flat.
Proof.(1) ⇒ (2) Let M be a left R-module.Then M has an almost F-injective cover ϕ : C → M by Theorem 2.9.On the other hand, there is an exact sequence A α → M → 0 with A free.Note that A is almost F-injective by ( 1), there exists a homomorphism β : A → C such that α = ϕβ.This follows that ϕ is epic.
(2) ⇒ (1) Let f : N → R R be an epic almost F-injective cover.Then the projectivity of R R implies that R R is isomorphic to a direct summand of N, and so R R is almost F-injective.
(1) ⇒ (3) Let M be any right R-module.Then M has an almost flat preenvelope f : M → F by Theorem 2.9(2).Since ( R R) + is a cogenerator, there exists an exact sequence , by Theorem 2.7(1) and Theorem 2.6, ( R R) + is almost flat, and so there exists a right R-homomorphism h : F → ( R R) + such that g = h f , which shows that f is monic.
(3) ⇒ (4) Assume (3).Then for every injective right R-module E, E has a monic almost flat preenvelope F, so E is isomorphic to a direct summand of F, and thus E is almost flat.Theorem 3.2.The following statements are equivalent for a ring R.
(2) pd R (R/I) ≤ 1 for every finitely presented left ideal I of R.
(3) Every submodule of an almost flat right R-modules is almost flat.
(4) Every finitely generated right ideal of R is almost flat.
(5) Every quotient module of an almost F-injective left R-module is almost F-injective.
almost F-injective modules and almost flat modules respectively, and we shall give some characterizations and properties of almost F-injective modules and almost flat modules.Moreover, we shall call a ring R left almost semihereditary if every finitely presented left ideal of R is projective.And we shall call a ring R left almost regular if every finitely presented left ideal of R is a direct summand.Left coherent rings, left almost semihereditary rings and left almost regular rings will be characterized by almost F-injective left R-modules and almost flat right R-modules.

( 7 )
There exists an almost pure exact sequence 0 → M → M → M → 0 of left R-modules with M injective.

( 8 )
⇒ (2) By (8), we have an almost pure exact sequence 0 → M → M f → M → 0 of left R-modules where M is almost F-injective, and so, for every finitely presented left ideal I, we have an exact sequence Hom(R/I, M )

Proposition 2 . 3 .
and (2) follows.The class of almost F-injective left R-modules is closed under direct limits and almost pure submodules.Proof.It is easy to see that the class of almost F-injective left R-modules is closed under direct limits by [1, Lemma 2.9(2)] and Theorem 2.2(2).Now let A be an almost pure submodule of an almost F-injective left R-module B. For any finitely presented left ideal I, we have an exact sequence Hom(R/I, B) → Hom(R/I, B/A) → Ext 1 (R/I, A) → Ext 1 (R/I, B) = 0. Since A is almost pure in B, the sequence Hom(R/I, B) → Hom(R/I, B/A) → 0 is exact.Hence Ext 1 (R/I, A) = 0, and so A is almost F-injective.Proposition 2.4.Let {M i | i ∈ I} be a family of left R-modules.Then the following statements are equivalent.

( 3 )
By Proposition 2.3 and Proposition 2.4, we need only to prove that the class of almost F-injective left R-modules is closed under pure quotients.Let 0 → A → B → C → 0 be a pure exact sequence of left R-modules with B almost F-injective.Then we get the split exact sequence 0 → C + → B + → A + → 0. Since B + is almost flat by (1), C + is also almost flat, and so C is almost F-injective by (1) again.(4) By Proposition 2.6, the class of almost flat right R-modules is closed under direct sums, direct summands and direct products.Let 0 → A → B → C → 0 be a pure exact sequence of right R-modules with B almost flat.Since B + is almost F-injective by (2), A + and C + are also almost F-injective, and so A and C are almost flat by (2) again.So the class of almost flat right R-modules is closed under pure submodules and pure quotients.

( 3 )
If A → B is an almost F-injective (resp.almost flat) preenvelope of a left (resp.right) R-module A, then B + → A + is an almost flat (resp.almost F-injective) precover of A + .Proof.(1) Since the class of almost F-injective left R-modules is closed under direct sums and pure quotients by Theorem 2.7(3), every left R-module has an almost F-injective cover by [5, Theorem 2.5].Since the class of almost F-injective left R-modules is closed under direct summands, direct products and pure submodules by Theorem 2.7(3), every left Rmodule has an almost F-injective preenvelope by [8, Corollary 3.5(c)].
) Let A → B be an almost F-injective preenvelope of a left R-module A. Then B + is almost flat by Theorem 2.7(1).For any almost flat right R-module M, M + is an almost F-injective left R-module by Theorem 2.7(2), and so Hom(B, M + ) → Hom(A, M + ) is epic.

( 7 )( 8 )( 9 )( 10 )
Every quotient module of an injective left R-module is almost F-injective.Every left R-module has a monic almost F-injective cover.Every right R-module has an epic almost flat envelope.For every left R-module A, the sum of an arbitrary family of almost F-injective submodules of A is almost F-injective.Proof.(1) ⇔ (2) It follows from the exact sequence 0 = Ext 1 (R, M) → Ext 1 (I, M) → Ext 2 (R/I, M) → Ext 2 (R, M) = 0. (1) ⇒ (3) Let A be a submodule of an almost flat right R-module B. Then for any finitely presented left ideal I, by (1), I is projective and hence flat.So the exactness of the sequence 0 = Tor 2 (B/A, R) → Tor 2 (B/A, R/I) → Tor 1 (B/A, I) = 0 implies that Tor 2 (B/A, R/I) = 0.And thus from the exactness of the sequence 0 = Tor 2 (B/A, R/I) → Tor 1 (A, R/I) → Tor 1 (B, R/I) = 0 we have Tor 1 (A, R/I) = 0, this follows that A is almost flat.

( 4 )( 2 )( 7 )
⇒ (1) Let I be a finitely presented left ideal.Then for any finitely generated right ideal K, the exact sequence 0 → K → R → R/K → 0 implies the exact sequence0 → Tor 2 (R/K, R/I) → Tor 1 (K, R/I) = 0 since K is almost flat.So Tor 2 (R/K, R/I) = 0,and then we obtain an exact sequence 0 = Tor 2 (R/K, R/I) → Tor 1 (R/K, I) → 0. Thus, Tor 1 (R/K, I) = 0, and so I is a finitely presented flat left R-module.Therefore, I is projective.⇒ (5) Let M be an almost F-injective left R-module and N a submodule of M. Then for any finitely presented left ideal I, by (2), Ext 2 (R/I, N) = 0. Thus the exact sequence0 = Ext 1 (R/I, M) → Ext 1 (R/I, M/N) → Ext 2 (R/I, N) = 0 implies that Ext 1 (R/I, M/N) = 0. Consequently, M/N is almost F-injective.123⇒ (1) Let I be a finitely presented left ideal.Then for any left R-module M, by hypothesis, E(M)/M is almost F-injective, and so Ext 1 (R/I, E(M)/M) = 0. Thus, the exactness of the sequence 0 = Ext 1 (R/I, E(M)/M) → Ext 2 (R/I, M) → Ext 2 (R/I, E(M)) = 0 implies that Ext 2 (R/I, M) = 0.And so, the exactness of the sequence 0 = Ext 1 (R, M) → Ext 1 (I, M) → Ext 2 (R/I, M) = 0 implies that Ext 1 (I, M) = 0, this shows that I is projective, as required.(6) ⇒ (8) Let M be a left R-module.Then by Theorem 2.9(1), there is an almost F-injective cover f : E → M. Since im( f ) is almost F-injective by (6), and f : E → M is an almost F-injective precover, for the inclusion map i : im( f ) → M, there is ahomomorphism g : im( f ) → E such that i = f g.Hence f = f (g f ).Observing that f : E → M is an almost F-injective cover and g f is an endomorphism of E, so g f is an automorphism of E, and hence f : E → M is a monic almost F-injective cover.(8) ⇒ (6) Let M be an almost F-injective left R-module and N be a submodule of M. By (8), M/N has a monic almost F-injective cover f : E → M/N.Let π : M → M/N be the natural epimorphism.Then there exists a homomorphism g : M → E such that π = f g.Thus f is an isomorphism, and whence M/N E is almost F-injective.

( 3 )
⇒ (9) Let M be a right R-module and Let {K i } i∈I be the family of all submodules of M such that M/K i is almost flat.Let F = M/ ∩ i∈I K i and π be the natural epimorphism ofM to F. Define α : F → i∈I M/K i by α(m + ∩ i∈I K i ) = (m + K i ) for m ∈ M,then α is a monomorphism.By Theorem 2.7(4), i∈I M/K i is almost flat.By (3), we have that F is almost flat.For any almost flat right R-module F and any homomorphism f : M → F , since M/Ker( f ) Im( f ) ⊆ F , M/Ker( f ) is almost flat by (3), and thus Ker( f ) = K j for some j ∈ I. Now we define g : F → F ; x + ∩ i∈I K i → f (x), then g is a homomorphism such that f = gπ.Hence, π is an almost flat preenvelope of M. Note that epic preenvelope is an envelope, so π : M → F is an epic almost flat envelope of M.

( 9 )
⇒ (3) Assume(9).Then for any submodule K of an almost flat right R-module M, K has an epic almost flat envelope ϕ : K → F. So there exists a homomorphism h : F → M such that i = hϕ, where i : K → M is the inclusion map.Thus ϕ is monic, and hence K F is almost flat.(6) ⇒ (10) Let A be a left R-module and {A γ | γ ∈ Γ} be an arbitrary family of almost Finjective submodules of A .Since the direct sum of almost F-injective modules is almost F-injective and γ∈Γ A γ is a homomorphic image of ⊕ γ∈Γ A γ , by (6), γ∈Γ A γ is almost F-injective.