The New Results in n -injective Modules and n -projective Modules

In this paper, we introduce and clarify a new presentation between the n -exact sequence and the n -injective module and n -projective module. Also, we obtain some new results about them.


Introduction
Category theory formalizes mathematical structures and their concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and their edges called arrows (or morphisms). This category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. The language of category theory has been employed to formalize concepts of other high-level abstractions such as sets, rings, and groups. Several terms were utilized in category theory, including theâmorphismâ that is used differently from their usage in the rest of mathematics. In category theory, morphisms obey specific conditions of theory. Samuel Eilenberg and Saunders Mac Lane introduced the concepts of categories, functors, and natural transformations The New Results in n-injective Modules and n-projective Modules 273 contained in these complexes and present it in the form of homological invariants of rings, modules, topological spaces, and otherâtangibleâ mathematical objects. A powerful tool for doing this is provided by spectral sequences. From its very origins, homological algebra has played an enormous role in algebraic topology. Its sphere of influence has gradually expanded and presently includes commutative algebra, algebraic geometry, algebraic number theory, representation theory, mathematical physics, operator algebras, complex analysis, and the theory of partial differential equations. K-theory is an independent discipline that draws upon methods of homological algebra, as does the noncommutative geometry of Alain Connes. This paper is organized as follows.
In this paper, we show to prove the important theorems of n-injective modules and n-projective modules. Finally, we recall the definition of n-projective module, and we give an open problem about some theorems of n-projective modules.

Preliminaries
All rings R in this paper are assumed to have an identity element 1 (or unit) (where r1 = r = 1r for all r ∈ R). We do not insist that 1 = 0; however, should 1 = 0, then R is the zero ring having only one element.
In this section, we recall some of the fundamental concepts and definitions, which are necessary for this paper. For details, we refer to [4,6,7,9,10,11]. Definition 2.1. An R-module M is injective provided that for every R-monomorphism g : A −→ B between R-modules, any R-homomorphism f : A −→ M can be extended to an R-homomorphism h : B −→ M such that hg = f ; i.e., the following diagram commutes 2. An R-module P is projective provided that for every R-epimorphism g : A −→ B between R-modules and R-homomorphism f : P −→ B, there exists an R-homomorphism f : P −→ B, there exists an R-homomorphism h : P −→ A such that gh = f ; i.e., the following diagram commutes We call B a basis of F .
Definition 2.4. Let M be an R-module. An element m ∈ M is divisible provided that for any r ∈ R that is not a righ zero-divisor, there exists an x ∈ M such that m = rx. We also say that M is a divisible module module provided that every element of M is divisible. Note that a divisible group is a divisible Z-module.
Definition 2.5. Let C be an additive category and f : A −→ B a morphism in C. A weak cokernel of f is a morphism g : B −→ C such that for all C ∈ C the sequence of abelian groups Definition 2.6. A category C is abelian if 1. C has a zero object.
2. For every pair of objects there is a product and a sum.
3. C Every map has a kernel and cokernel.
4. C Every monomorphism is a kernel of a map.
5. C Every epimorphism is a cokernel of a map. 2. the distributive laws hold: given morphisms 3. C has a zero object.
4. C has finite product and finite coproduct.
8. An abelian group D is said to be divisible if given any y ∈ D and 0 = n ∈ Z, there exists x ∈ D such that nx = y. ...
is exact if and only if f is an R-monomorphism, g is an R-epimorphism, and This type of sequence is called short exact.
Definition 2.11. Let C be an additive category and f : A −→ B a morphism in C. A weak cokernel of f is a morphism g : B −→ C such that for all C ∈ C the sequence of abelian groups These properties are subsumed in the following commutative diagram: Clearly, a weak cokernel g of f is a cokernel of f if and only if g is an epimorphism. The concept of weak kernel is defined dually.
The subcategory X is said to be a covariantly finite subcategory of C if any object A of C has a left X -approximation. We can defined X -epic morphism, right X -approximation and contravariantly finite subcategory dually. The subcategory X is called functorially finite if it is both contravariantly finite and covariantly finite.
Definition 2.13. Let C be an additive category and d 0 : such that, for all Y ∈ C the induced sequence of abelian groups In this case, we say the sequence Remark 2.14. When we say n-cokernel we always means that n is a positive integer. We note that the notion of 1-cokernel is the same as cokernel. we can define n-kernel and left n-exact sequence dually.
Definition 2.15. Let C be an additive category. An n-exact sequence in C is a complex in Ch n (C) such that (d 0 , ..., d n−1 ) is an n-ker of d n , and (d 1 , ..., d n ) is an n-coker of d 0 . The sequence (3.1) is called n-exact if it is both right n-exact and left n-exact.
Theorem 2.16. Let A, B, {B i |i ∈ I}, {A j |j ∈ J, Jisf inite} be modules over a ring R . Then there is isomorphisms of abelian groups: is also a short exact sequence, where Ψ(f ) = f ψ and Θ(f ) = f θ.
Proposition 2.22. Let R be a ring. A direct sum of R-modules i∈I P i is projective if only if each P i is projective.

Q is an injective Z-module by Lemma (2.26) since for every
Z-homomorphism f : nZ −→ Q, where nZ is an ideal of Z for 0 = n ∈ Z, there exists a Z-homomorphism g : Z −→ Q defined by g(z) = zf (n) n , so g(nz) = (nz)f (n) n = zf (n) = f (nz) for every nz ∈ Z.

n-injective Module
Definition 3.1. Let C be an category of R-modules, X i ∈ obj(C) for all 0 ≤ i ≤ n, if there is M ∈ C the induced sequence of abelian groups is right n-exact.
Proposition 3.2. Let C be an category of R-modules, X i ∈ obj(C) for all 0 ≤ i ≤ n, and d i for all 0 ≤ i ≤ n − 1 is a morphism in C. A direct product of R-modules i∈I J i is n-injective if only if J i is n-injective for every i ∈ I.
Proof. Let C be an category of R-modules, X i ∈ obj(C) for all 0 ≤ i ≤ n, and d i for all 0 ≤ i ≤ n − 1 is a morphism in C. The sequence of R-module in C Suppose that i∈I J i is n-injective. To show that, J i is n-injective for each i ∈ I. Now if there is i∈I J i the induced sequence of abelian groups this sequence is is right n-exact. By Theorem 2.16, (1), for each i ∈ I. Then this sequence Conversely, suppose that J i is n-injective. To show that, i∈I J i is n-injective for each i ∈ I. Now if there is J i the induced sequence of abelian groups this sequence is is right n-exact. By Theorem 2.16, (1). Then this sequence is right n-exact. Then i∈I J i is also n-injective. Corollary 3.3. Let C be an category of R-modules, X i ∈ obj(C) for all 0 ≤ i ≤ n, and d i for all 0 ≤ i ≤ n − 1 is a morphism in C. Let R be an integral domain and let K the field of fractions of R. Then K is an n-injective R-module.
Proof. By Corollary 2.19, k is injective R-module. Let C be an category of R-modules, X i ∈ obj(C) for all 0 ≤ i ≤ n, and d i for all 0 ≤ i ≤ n − 1 is a morphism in C. The sequence of R-module in C is left n-exact. By Theorem 2.21 is right n-exact. Then K is n-injective module.
Proof. Let C be an category of R-modules, X i ∈ obj(C) for all 0 ≤ i ≤ n, and d i for all 0 ≤ i ≤ n − 1 is a morphism in C. The sequence of R-module in C is left n-exact.
Suppose that M λ is n-injective. To show that, ⊕ λ∈Λ M λ is n-injective for each λ ∈ Λ. Now if there is M λ the induced sequence of abelian groups this sequence is is right n-exact. If Λ is finite by Theorem 2.17, for every λ ∈ Λ, Then this sequence is right n-exact. Then ⊕ λ∈Λ M λ is also n-injective.
Proposition 3.5. Every R-module injective is not n-injective.
Proof. Let {M λ } λ∈Λ be a family of R-modules. If ⊕ λ∈Λ M λ is injective, then M λ is injective for every λ ∈ Λ but ⊕ λ∈Λ M λ is not n-injective for every λ ∈ Λ and then, M λ is n-injective for every λ ∈ Λ. Definition 3.6. Let C be an category of R-modules, Y i ∈ obj(C) for all 0 ≤ i ≤ n + 1, and f i for all 0 ≤ i ≤ n is a morphism in C. An R-module P is n-projective if the sequence of R-module in C is rightt n-exact if there is P ∈ C the induced sequence of abelian groups Proposition 3.7. Let C be an category of R-modules, Y i ∈ obj(C) for all 0 ≤ i ≤ n + 1, and f i for all 0 ≤ i ≤ n is a morphism in C. A direct sum of R-modules ⊕ i∈I P i is n-projective if only if P i is n-projective for every i ∈ I and I is finite.
Proof. Let C be an category of R-modules, Y i ∈ obj(C) for all 0 ≤ i ≤ n + 1, and f i for all 0 ≤ i ≤ n − 1 is a morphism in C. The sequence of R-module in C is right n-exact. Suppose that ⊕ i∈I P i is n-projective. To show that, P i is n-projective for each i ∈ I. Now if there is ⊕ i∈I P i the induced sequence of abelian groups this sequence is is left n-exact. If I is finite by Theorem 2.16, (2) Hom C (⊕ i∈I P i , Y j ) ∼ = ⊕ i∈I Hom C (P i , Y j ) for every i ∈ I, Then this sequence is left n-exact. Then P i is n-projective for each i ∈ I.
Conversely, suppose that P i is n-projective. To show that, ⊕ i∈I P i is n-projective for every i ∈ I and I is finite. Now if there is P i the induced sequence of abelian groups this sequence is is left n-exact. If I is finite by Theorem 2.16, (2) Hom C (⊕ i∈I P i , Y j ) ∼ = ⊕ i∈I Hom C (P i , Y j ) for every i ∈ I. Then this sequence ...f n−1 −→ Hom C (⊕ i∈I P i , Y n )f n −→ Hom C (⊕ i∈I P i , Y n+1 ) −→ 0 is left n-exact. Then ⊕ i∈I P i is also n-projective.