The New Results in Injective Modules

In this paper, we introduce and clarify a new presentation between the divisible module and the injective module. Also, we obtain some new results about them.


Introduction
In mathematics, a module is one of the fundamental algebraic structures used in abstract algebra. A module taking its scalars from a ring R is called an R-module. Thus, a module, like a vector space, is an additive abelian group, a product is defined between elements of the rings and elements of the module that is distributive over the addition operation of each parameter and is compatible with the ring multiplication. In mathematics, especially in the area of abstract of algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers. Injective modules have been heavily studied, and a variety of additional notions are defined in terms of them. Injective co generators are injective modules that faithfully represent the entire category of modules. Injective resolutions measure how far from injective a module is terms of the injective dimension and represent modules in the derived category. Injective hulls are maximal essential extensions, and turn out to be minimal injective extensions. Over a Noetherian ring, every injective modules is uniquely a direct sum of incomparable modules, and their structure is well understood. An injective module over one ring, may not be injective over another, but there well-understood methods of changing rings which handle special cases. Rings which are themselves injective modules have a number of interesting properties and include rings such as group rings of finite groups over fields. Injective modules include divisible groups and are generalized by the notion of injective objects in category theory. In mathematics, one can often define a direct Product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one talks a bout the product in category theory, which formalizes these notions. Examples are the product of sets, groups (describe below), rings and other algebraic structures. The product of topological spaces is another instance. There is also the direct sum in some areas this is used interchangeably, while in others it is a different concept.
In this paper, we show to prove the important theorems and examples of injective modules. Next, we show a strong relationship between the injective module and the divisible module, such that every injective module gives divisible but the converse needs another condition P.I.D. Finally, we recall the definition of n-cokernel, n-kernel, and n-exact sequence and we give an open problem about n-injective modules.

Preliminaries
In this section, we recall some of the fundamental concepts and definitions, which are necessary for this paper. For details we refer readers to [1,[6][7][8] Definition 2.1. Let R be a ring. A (left) R-module is an additive abelian group A together with a function R × A −→ A (the image of (r, a) being denoted by ra) such that for all r, s ∈ R and a, b ∈ A: (i) r(a + b) = ra + rb.
If R has an identity element 1 R and (iv) 1 R a = a for all a ∈ A, then A is said to be a unitary R-module. If R is a division ring, then a unitary R-module is called a (left) vector space.
Definition 2.2. Suppose R is a ring, and {M i } i∈I . The final family are R-modules. We define the direct product of modules M i , denoted by Scalar addition and multiplication is defined by Definition 2.3. Suppose R is a ring, and A and B are R-modules. We define the direct sum of modules A and B as where all algebraic operations are defined componentwise. In particular, suppose that A and B are left R-modules, then and r(a b) = (ra rb). Definition 2.5. Let M i∈Z be a family of R-modules, and let f ii∈Z be a family of and Im(f ) = Ker(g). This type of sequence is called short exact.
Definition 2.6. 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 An element m ∈ M is divisible provided that for any r ∈ R that is not a right zero-divisor, there exists an x ∈ M such that m = rx. We also say that M is a divisible module provided that every element of M is divisible. Note that a divisible group is a divisible Z-module.
Definition 2.8. 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 is exact. Equivalently, g is a weak cokernel of f if f g = 0 and for each morphism h : B −→ C such that f h = 0 there exists a (not necessarily unique) morphism p : C −→ C such that h = gp. 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.  ii. There is an R-homomorphism h : iii. There is an R-homomorphism k : B −→ A 1 with kf = 1 A 2 ;

Main Results
Definition 3.1. An R-module J is an injective module if J satisfies one of the equivalent conditions of Proposition (2.9) ...). Now it is very easy to check that, hg = f .
Conversely, suppose that i∈I J i is injective. To show that, J i is injective for each i ∈ I. Now in this diagram We have to find h i . Define h i : B −→ J i to be h i = π i h. Now it is very easy to check that, h i g = f ∀i ∈ I. Here, h i g = π i hg = π i i i f = IJ i f = f ∀i ∈ I. If H = B and b ∈ B − H, then L = {r ∈ |rb ∈ H} is left ideal of R. The map L −→ J given by r −→ h(rb) is a well-defined R-module homomorphism. By the hypothesis there is a R-module homomorphism k : R −→ J such that k(r) = h(rb) for all r ∈ L. Let c ∈ k(1 R ) and define a maph : H + Rb −→ J by a + rb −→ h(a) + rc. We claim thath is well-defined. For if a 1 + r 1 b = a 2 + r 2 b ∈ H + Rb , then a 1 − a 2 = (r 2 − r 1 )b ∈ H ∩ Rb. Hence r 2 − r 1 ∈ L and h(a 1 )−h(a 2 ) = h(a 1 −a 2 ) = h((r 2 −r 1 )b) = k(r 2 −r 1 ) = (r 2 −r 1 )k(1 R ) = (r 2 −r 1 )c. Therefore,h : H + Rb −→ J is an R-module homomorphism that is an element of the set S. This contradicts the maximality of h since b / ∈ H and hence H H+Rb. Therefore, H = B and J is injective.

Q is an injective Z-module by Lemma (3.3) 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.
Secondly, we will see that Θ is an R-epimorphism. Let g ∈ Hom R (A, M ). Since M is injective, there exists an f ∈ Hom R (B, M ) such that g = f θ = Θ(f ). Thus, Θ is an R-epimorphism.
Lemma 3.13. Every abelian group A may be embedded in a divisible abelian group.
Proof. There is a free Z-module F and an epimorphism F −→ A with kerK so that F/K ∼ = A. Since F is a direct sum of copies of Z and Z ⊂ Q, F may be embedded in a direct sum D of copies of the rationals Q. But D is a divisible group by Proposition (3.2) and Lemma (3.11). If f : F −→ D is the embedding monomorphism, then f induces an isomorphism But D/f (K) is divisible since it is the homomorphic image of a divisible group.
Lemma 3.14. If J is a divisible abelian group and R is a ring with identity, then Hom Z (R, J) is an injective left R-module. Proof. Since A is an abelian group, there is a divisible group J and a group monomorphism f : A −→ J by Lemma (3.13). The map f : Hom Z (R, A) −→ Hom Z (R, J) given on g ∈ Hom Z (R, A) byf (g) = f g ∈ Hom Z (R, J) is easily seen to be an R-module monomorphism. Since every R-module homomorphism is a Z-module homomorphism, we have Hom R (R, A) ⊂ Hom Z (R, A). In fact, it is easy to see that Hom R (R, A) is an R-submodule of Hom Z (R, A). Finally, the map A −→ Hom R (R, A) given by a → f a , where f a (r) = ra, is an R-module monomorphism (in fact it is an isomorphism). Composing these maps yields an R-module monomorphism Since Hom Z (R, J) is an injective R-module by Lemma (3.14), we have embedded A in an injective.