In this paper all rings are right ring and all mods are right mod X over a ring R. Baer’s Criteria study Inj mods in certain rings called a projective(Proj) mod [1]. Trlifaj [2] say that perfect (Pfc) rings are testing and Trlifaj [3] get some examples(Exs) of this ring. Faith asked two questions one Alhilali et al. [4] are rings does the dual B.Criterion hold and in Trlifaj [5] asked when does R-Proj imply Proj for all mods? this questions ensured by Trlifaj [3], Alhilali et al. [4] and Faith [6]. Trlifaj [3] testing rings is the ring whose R-Proj mods are Proj [2]. It was shown that Pfc rings are testing and Trlifaj [3] introduce Exs of local rings that are not testing. Yusuf and Engin [1] introduce a mod X is mx-Proj. Generalized mx-Proj mod by Dinh et al. [7]. Nicholson and Yousif [8] introduced a dualization of the notion of non-inj rings and mods and a generalization by Trlifaj [2], Trlifaj [3], Nicholson and Yousif and Amini et al. [9]. The rings whose flat mods are R-Proj and mx-Proj are characterized in Nicholson and Yousif [8], Amini et al. [9] and Amini et al. [10] resp.

In this work we generalization of mx-Proj mod that a mod X is said to be a Ner.Mx.Pro.Mod if for every Epi θ:R→R/E where E is a nearly mx ideal of R and ∀ Hom Ψ:X →R/E, ∃ a Hom ɸ:X→R s.t. θɸ=Ψ.

This paper contains four sections, in section (2); we introduce some results of Ner.Mx.Pro.Mods are investigated. We obtain that R-Pro and Ner.Mx.Pro.Mod coincide over the ring of integers and over Pfc rings.

In section (3); submods of Ner.Mx.Pro.Mods are Ner.Mx.Pro.Mods, introduce singular and simple mods are Inj. Pure submods of Ner.Mx.Pro.Mods are Ner.Mx.Pro.Mods, over commutative rings and over semi-local rings.

Finilly, in section (4) we study a characterizations of semi-Pfc, Pfc and F.G. rings in terms of Ner.Mx.Proj.Mods are given. Every Ner.Mx.Pro.Mod mod of finite length (F.L) is Pfc [1]. Any finitely generated (F.G) R-Proj mod is Proj. But it is not true if we replaced Proj by Ner.Mx.Pro.Mod. also we can get F.G.Ner.Mx.Pro.Mod are Proj when we put some condition. We prove that, Ris semi-local and Ner.Mx.Pro.Mods are non-singular if and only if R is non-singular and Pfc.

Nearly Maximal Projective Modules

In this section, we introduce a definition of Ner.Mx.Pro.Mod with some properties and characterizations.

Definition (2.1)

A mod X is said to be Ner.Mx.Pro.Mod if ∀ Epi θ:R→R/N, with N is a nearly mx ideal of R and ∀ Hom Ψ:X→R/N, ∃ a Hom ɸ:X→R s.t θɸ=Ψ. 

Remarks and Examples (2-2)

• Each Proj mod is Ner.Mx.Pro.Mod

• The converse of the previous remark is not true in general, as we see in the following example

• If X is a mod over a ring R, with Rad(X) = X is Ner.Mx.Pro.Mod, since X has no simple, factors, Z-mod Q is Ner.Mx.Pro.Mod but it is not Proj mod

• If X is a simple and Ner.Mx.Pro.Mod, then it is Proj

If X is a simple mod and 1_X:X→X is the identity function, then by Ner.Mx.Pro.Mod of X there is a Hom θ:X → R s.t πθ = 1_X, where π:R→X/N is a natural Epi. Then R≅N⊕X and so X is Proj mod.

Corollary (2-3)

If X is a mod over a ring R. Then the following statements are equivalent:

• R is semi-simple ring

• ∀ mod X is Ner.Mx.Pro.Mod

• ∀ F.G mod X is Ner.Mx.Pro.Mod 

• ∀ cyclic mod is Ner.Mx.Pro.Mod

• ∀ simple mod is Ner.Mx.Pro.Mod

The following propositions(props) give some conditions to get remark (1-2) (2) is true.

The proof of the following prop is similar to Büyükasık [12], hence we omitted. 

Proposition (2-4)

N_i is a Ner.Mx.Pro.Mod ∀ iϵI iff ⊕_iϵI N_i is Ner.Mx.Pro.Mod. 

Proposition (2-5)

and X be a Ner.Mx.Pro.Mod, then X₁ and X₂ are Ner.Mx.Pro.Mods.

Proof

If X is a Ner.Mx.X-subPro.Mod, ∀ Hom θ:X→X₁ and ∀ Epi Ψ:X₂→X₁, ∃a Hom ɸ:X→X₂ s.t Ψɸ = θ. Then X is Proj relative to both X₁ and X₂.

Proposition (2-6)

If X is a mod, then the following statements are equivalent:

• X is Ner.Mx.Pro.Mod

• X is Ner.Mx.N-subproj.Mod ∀simple mod N .

• ∀Epi θ:L→N with N is a simple and Hom ɸ:X→N, ∃a Hom Ψ:X→L s.t. θΨ=ɸ

Proof

1)⇒3)Let θ:L→N be an Epi with N is simple mod and ɸ:X→N is a Hom. Since N is simple, ∃ an Epi π:R→N. By the hypothesis ∃a Hom Ψ:X→R s.t π Ψ = ɸ. Since R is Proj, ∃ a Hom f:R→L s.t θf`= π. Then θ(f f`)=πf= ɸ. 

3)⇒1)it is clear. And by def. we can get 2)⇔3).

Proposition (2-7)

If X is a Ner.Mx.X₁-sub-Pro.Mod and Ner.Mx.X₂-sub-Pro.Mod, then X is Ner.Mx.X`-sub-Pro.Mod.

Proof

Let θ:L→X` be an Epi with L is Ner.Mx.Pro.Mod. Then using the pullback diagram of θ:L→X` and Ψ:X₁ →X` and applying Hom(X,−), we get the following commutative(com) diagram with exact rows and columns.

Since X is Ner.Mx.X₁-sub-Pro.Mod and Ner.Mx.X₂-sub-Pro.Mod, π and φ are Epi, hence θ is Epi [12].              

Proposition (2-8)

X is Ner.Mx.Pro.Mod if and only if X is Ner.Mx.L-sub-Pro.Mod for any mod L with composition(comp) length cl(L) <∞.

Proof

Let X be a Ner.Mx.Pro.Mod and L be a mod with cl(L) = n, where nϵZ⁺. Then ∃ a comp series 0 = A₀ ⊂ A₁ ⊂ ··· ⊂ A_n = L with each comp factor A_i+1/A_i simple. Consider 0→A₁ → A₂→A₂/A₁→0 is a short exact sequence. Since X is Ner.Mx.Pro.Mod, then X is Ner.Mx.X₁-sub-Pro.Mod and Ner.Mx.A₂/A₁-sub-Pro.Mod (prop 2-6). Also by (prop 2-4), X is Ner.Mx.X₁-sub-Pro.Mod. By the same way we can get X is Ner.Mx.X_i-sub-Pro.Mod ∀0≤i≤ n. Hence, X is Ner.Mx.N-sub-Pro.Mod. 

Conversely; all simple mod has finitely length(F.L), X is Ner.Mx.Pro.Mod (prop 2-6). 

Corollary (2-9)

An Z-mod X is Ner.Mx.Pro.Mod iff X is Z-Proj.

Proof

Cyclic Z-mod X is isomorphism(Iso) to integer numbers or to a F.direct sum of Z-mods of Fuchs [13] and by prop (2-8) proof is hold.

Corollary (2-10)

Let X be F. Lcomp mod. If X is Ner.Mx.Pro.Mod, then it is a Proj.

Proof

Let θ:Rⁿ →X be an Epi. The mod X is Ner.Mx.X-sub-Pro.Mod (prop 2-8) ∃ a Hom Ψ:X→Rⁿ s.t 1_X = θΨ. Therefore the function θ is split and X is a Proj.

Remark (2-11)

It is not necessary that submods of Ner.Mx.Pro.Mod is Ner.Mx.Pro.Mod, as we see in the following example.

Example (2-12)

Consider the ring R = Z/p²Z, for some p is prime integer. R is Ner.Mx.Pro.Mod, where the simple ideal _pR is not Ner.Mx.Pro.Mod, since the Epi R→_pR→0 is not split.

Nearly Maximal Projective Modules over SV-Ring

Throughout this part R is a ring and X is a singular and simple mod, we start this section by the following definition, which appeared in Yusuf and Engin [1].

Definition (3-1)

A ring R is called SV-ring if ∀ singular simple mods over R are Inj. 

Proposition (3-2)

Consider the following conditions for self-Inj ring R, then the following statements are equivalent:

• R is SV-ring

• Submods of Ner.Mx.Pro.Mod are Ner.Mx.Pro.Mod

• Submods of Proj mods are Ner.Mx.Pro.Mod

• ∀ ideal E of R is Ner.Mx.Pro.Mod

Proof

1)⇒2) Let A be a submod of a Ner.Mx.Pro.Mod X. Consider the following diagram:

where, B is a simple mod, I:A→X is the inclusion function and π:R→B is a canonical quotient function. B is simple mod, so it is Proj or singular, a former implies π:R→B splits, ∃ a Hom ε:B →R s.t επ =1_R. B is singular, so it is Inj by the hypothesis. Thus there is a Hom ɸ:X→B s.t ɸI = θ. Since X is Ner.Mx.Pro.Mod, there is a Hom Ψ : X → R s.t πΨ = ɸ. Hence, π(ΨI) = ɸI = θ. ∃ a Hom from A to R, thus the diagram is commute and A is Ner.Mx.Pro.Mod: 

• 2)⇒3)⇒4)Clear

• 4)⇒1)Let P be an ideal of R and E is a mx ideal of R

Consider the following diagram:

where, R/E is a simple mod, I:P→R is the inclusion function and π:R →R/E is the canonical quotient function. Since P is Ner.Mx.Pro.Mod, ∃ a Hom h:E→R s.t πh=f. Since R is Inj, ∃ a Hom λ:R →R s.t λi = h. Now, β = πλ:R→R/J satisfies βi = πλi = πh = f, as required. 

Over com rings each simple mod is pure-Inj [14]. Hence we can get the following result.

Proposition (3-3)

Let R be a com ring and A be a pure submode of Ner.Mx.Pro.Mod X, then A is Ner.Mx.Pro.Mod.

Proof

Let X be a Ner.Mx.Pro.Mod and A is a pure submod of X. And B is a simple mod. θ:A→B be a Hom. Since B is pure-Inj and A is a pure submod of X, ∃ Ψ:X→B s.t ΨI= θ, where I:A→X is the inclusion function and since X is a Ner.Mx.Pro.Mod, ∃ a Hom ɸ:X→R s.t Ψ=πɸ , where π:R→B is the natural Epi, thus we get θ = ΨI = πɸI, i.e the function ɸI:A→R lifts θ. Therefore A is Ner.Mx.Pro.Mod.

Proposition (3-4)

Let R be a semi-local ring, A is a pure submods of Ner.Mx.Pro.Mod. X, then A is Ner.Mx.Pro.Mod.

The proof is similar to the previous prop, hence we omitted.

Proposition (3-5)

Let R be a ring and τ be a pre-radical with τ(R) = 0. If X is a Ner.Mx.Pro.Mod, then X/τ(X) is Ner.Mx.Pro.Mod.

Proof

Let X be a Ner.Mx.Pro.Mod and θ:X/τ(X)→B is a Hom where B is a simple mod. Now, consider .

Since X is Ner.Mx.Pro.Mod, ∃ a Hom Ψ : X → R s.t θπ = ηΨ. Since Ψ(τ(X))⊆ τ(R) = 0, τ(X)) ⊆Ker(Ψ) and so ∃ a Hom ɸ:X/τ(X)→R s.t ɸπ = Ψ. Now, since ηɸπ = ηΨ = θπ and π is an Epi, ηɸ = θ and so X/τ(X) is Ner.Mx.Pro.Mod.

F.G Nearly Maximal Projective Modules

In this section, we will take F.G Ner.Mx.Pro.Mod, first we introduce the following remark.

Remark (4-1)

F.G Ner.Mx.Pro.Mod need not be Proj, as we see in the following example.

Example (4-2)

Let R be not semi-hereditary V-ring. Then R is non-F.G Proj ideal, say I (prop 32), the F.G ideal I is Ner.Mx.Pro.Mod but it is not Proj.

The following props over the hereditary, semi-Pfc and non-singular and self-Inj (resp) ring, F.G Ner.Mx.Pro.Mod is Proj.

Proposition (4-3)

If R is hereditary Noetherian ring and X is Ner.Mx.Pro.Mod. Then X is Proj.

Proof

Let X be a F.G Mod ∃ a simple mod A and a non-zero θ:X →A. By Ner.Mx.Pro.Mod, ∃ Ψ:X →R that is lift of θ, Ψ(X) is a Proj. Hence X=X₁ ⊕B, where X₁≅Ψ(X). Thus B is F.G Ner.Mx.Pro.Mod. 

By the same way, we get B = X₂ ⊕C, for some Proj submod X₂ of X. And X has F.uniform dimension. After finitely steps, we can get X = X₁⊕ ··· ⊕X_n, where X_i is a proj for each I = 1,...,n .

Proposition (4-4)

If R is a semi-Pfc ring and X is Ner.Mx.Pro.Mod. Then X is a Proj.

Proof

Since R is semi-local, X/Rad(X) is semi-simple. Thus X/Rad(X) has a Proj cover. 

Let θ:J →X/Rad(X) be a Proj cover, where J is F.G and hence X is J-Proj. Then the canonical function π:X →X/Rad(X) lifts to a Hom Ψ:X→J [15].

Ψ is an Epi. Thus Ψ splits and so X=C⊕Ker(Ψ) for some submod C of X, where C≅J.ker(Ψ) and K≅P. So ker(Ψ) ⊆Rad(X) ≪ X, hence X is Proj.

Proposition (4-5)

If R is a non-singular self-Inj ring and X is Ner.Mx.Pro.Mod. Then X is a Proj.

Proof

Let X be a F.G Ner.Max.Pro.Mod and R is a non-singular ring (prop 3-5), X/Z(X) is Ner.Mx.Pro.Mod. Since X/Z(X) is a F.G, ∃ an Epi θ:L→X/Z(X) s.t. L is a F.G free. Its means Ker(θ) is closed in L. By the Inj of L, ker(θ) is a direct summand of L and so X/Z(X) is Proj. Then X = Z(X)⊕B for some Proj submod B of X. We claim that Z(X) = 0, assume to the contrary that X/Z(X) = 0. Since Z(M) is a F.G submod of X, ∃ a non-zero Epi Ψ:Z(M)→A for some simple mod A. Then Z(X) is Ner.Mx.Pro.Mod(prop2-5), ∃ a non-zero Hom ɸ:Z(X)→R s.t πɸ = Ψ, where π:R →A is the natural Epi. But then ɸ(Z(X))⊆Z(R_R)=0 which is a C!, thus Z(X) = 0 and hence X is Proj.

Proposition (4-6)

Let X be a small radical mod over a semi-Pfc ring R. If X is Ner.Mx.Pro.Mod, then X is Proj.

Proof

Let X be a Ner.Mx.Pro.Mod with Rad(X)≪Xand θ:X→ is a Homo. Since A_i where A_i is simple ∀iϵI.

. Choose φ = π_i π, ∃ a Hom Ψ:X→R s.t ɸΨ = π_i θ. Since π_i is splis and is semi-simple ∃ a Hom μ_i: A_i→ A_i such that μ_i π_i = R/J(R). 

Then πΨ = μ_iɸΨ = μ_iπ_iθ = θ. Hence X is Proj [17].

Proposition (4-7)

A ring R is Pfc iff R is semi-local and every Ner.Mx.Pro.Mod is Proj.

Proof

Over a Pfc ring R ∀ mod has small radical, it follows from (prop4-6) that every Ner.Mx.Pro.Mod is Proj. 

Conversely; assume that R is semi-local and every Ner.Mx.Pro.Mod is Proj. Let X be a non-zero mod. Consider Rad(X) ≠ X and X has no mx submod, i.e., Rad(X) = X. Since Hom(X,A) = 0 ∀ simple mod A, X is Ner.Mx.Pro.Mod. Thus X is Proj, by the hypothesis. Since Proj mods have mx submods, this is a Cǃ. Hence every mod has a mx submod. Since R is semi-local, R is Pfc [12]. 

Recall that if R is a Pfc ring, then ∀ R-Proje mod is Proj. 

Corollary (4-8)

Let X be a mod over a Pfc ring R. Then the following statements are equivalent.

• X is Proj

• X is R-Proj

• X is Ner.Mx.Pro.Mod

Let A be a submod X is radical if A has no mx submods, i.e., A = Rad(A). If P(X) = ∑Rad(A). Then P(X) is the largest radical submod of X and so Rad(P(X)) = P(X). Moreover, P is an idempotent radical with P(X)⊆Rad(X) and P(X/P(X)) = 0.

Proposition (4-9)

Every Ner.Mx.Pro.Mod X is non-singular, then R is non-singular and mx ring.

Proof

Clearly the ring R is non-singular. If R is V-ring, then Rad(X) = 0 for each mod X. Thus R is a mx ring. Suppose R is not V-ring and let K be a non-Inj simple mod. We see E(K) has no non-zero radical submod i.e., P(E(K)) = 0. Suppose Rad(P(E(K))) = P(E(K))/K = 0. Then P(E(K))/K is singular, since Rad(P(E(K))/S) = P(E(K))/K, P(E(K))/K is Ner.Mx.Pro.Mod. Therefore for every simple mod A, P(E(K)) = 0. Let X be a non-zero mod. We claim that Rad(X)≠X. Assume to the contrary that Rad(X) = X. Let 0≠xϵX and xR be a mx submod of X, since A is small. Then the simple mod A = xR/K is non-Inj. Now the obvious function xR→E(A) extends to a non-zero function θ:X →E(A). Since P(Im(θ)) ⊆ P(E(A)) = 0, P(X/Ker(θ)) = 0, that’s Cǃ with P(X) = X. Hence Rad(X)≠X ∀X and so R is mx ring.

For semi-local rings we are able to see when all Ner.Mx.Pro.Mods are non-singular.

Corollary (4-10)

For a ring R, the following statements are equivalent:

• R is semi-local and each Ner.Mx.Pro.Mod is non-singular

• R is Pfc and non-singular

Faith [6] say that R is QF iff each Proj mod is Inj.

Proposition (4-11)

A ring R is QF iff each Ner.Mx.Pro.Mod is Inj.

Proof

Let X be a Ner.Mx.Pro.Mod, since Rad(X)≪X. Then by prop(4-6) it is Proj, so X is Inj.

Conversely; clear Trlifaj [3], By adapting to proof and choosing the ideal E in their proofs to be mx, we can establish the following result.

Proposition (4-12)

• If X is a mod s.t Ext¹_R(X,E) = 0, ∀ mx ideal E, then X is Ner.Mx.Pro.Mod

• If R is a self-Inj ring and X is Ner.Mx.Pro.Mod. Then Ext¹_R(X,E)=0 ∀ mx ideal E

The aim of this manuscript is to introduced a new generalized of Proj mod which is Ner.Mx.Pro.Mod that if ∀ Epi θ : R → R/E where E is a Ner.Mx. ideal of R and every Hom Ψ : X → R/E, ∃ a Hom ɸ : X → R s.t θɸ = Ψ. Some properties of Ner.Mx.Pro.Mods are investigated. we obtain that R-Proj and Ner.Mx.Pro.Mod coincide over the ring of integers and over Pfc rings. And submods of Ner.Mx.Pro.Mods are Ner.Mx.Pro.Mods, provided that singular and simple mods are Inj, pure submods of Ner.Mx.Pro.Mods are Ner.Mx.Pro.Mods, over com. rings and over semi-local rings.

1. Yusuf, A. and B. Engin. “Max-Projective Modules.” Journal of Algebra and Its Applications, 2020, article no. 2150095(25). https://doi.org/10.1142/S0219498821500 95X.

2. Trlifaj, J. “Whitehead Test Modules.” Transactions of the American Mathematical Society, vol. 348, 1996, pp. 1521-1554.

3. Trlifaj, J. “Faith’s Problem on R-Projectivity Is Undecidable.” Proceedings of the American Mathematical Society, vol. 147, no. 2, 2019, pp. 497-504.

4. Alhilali, H. et al. “When R Is a Testing Module for Projectivity?” Journal of Algebra, vol. 484, 2017, pp. 198-206.

5. Trlifaj, J. “Dual Baer Criterion for Non-Perfect Rings.” Forum Mathematicum, online publication, 2020. https:// doi.org/10.1515/forum-2019-0028.

6. Faith, C. Algebra II: Ring Theory. Springer-Verlag, 1976.

7. Dinh, H.Q. et al. “Quasi-Projective Modules over Prime Hereditary Noetherian V-Rings Are Projective or Injective.” Journal of Algebra, vol. 360, 2012, pp. 87-91.

8. Nicholson, W.K. and M.F. Yousif. “Mininjective Rings.” Journal of Algebra, vol. 187, no. 2, 1997, pp. 548-578.

9. Amini, B. et al. “Almost-Perfect Rings and Modules.” Communications in Algebra, vol. 37, no. 12, 2009, pp. 4227-4240.

10. Amini, A. et al. “Rings over Which Flat Covers of Finitely Generated Modules Are Projective.” Communications in Algebra, vol. 36, no. 8, 2008, pp. 2862-2871.

11. Büyükasık, E. “Rings over Which Flat Covers of Simple Modules Are Projective.” Journal of Algebra and Its Applications, vol. 11, no. 3, 2012, article no. 1250046.

12. Anderson, F.W. and K.R. Fuller. Rings and Categories of Modules. Springer-Verlag, 1992.

13. Fuchs, L. Infinite Abelian Groups. Academic Press, 1970.

14. Cheatham, T.J. and J.R. Smith. “Regular and Semisimple Modules.” Pacific Journal of Mathematics, vol. 65, no. 2, 1976, pp. 315-323.

15. Wisbauer, R. Foundations of Module and Ring Theory. Gordon and Breach Science Publishers, 1991.

16. Ketkar, R.D. and N. Vanaja. “R-Projective Modules over a Semiperfect Ring.” Canadian Mathematical Bulletin, vol. 24, 1981, pp. 365-367.

17. Büyükasık, E. et al. “Rad-Supplemented Modules.” Rendiconti del Seminario Matematico della Università di Padova, vol. 124, 2010, pp. 157-177.