# Difference between revisions of "Injective module"

An injective object in the category of (right) modules over an associative ring with identity $R$, i.e. an $R$- module $E$ such that for any $R$- modules $M$, $N$, for any monomorphism $i : N \rightarrow M$, and for any homomorphism $f : N \rightarrow E$ there is a homomorphism $g : M \rightarrow E$ that makes the following diagram commutative

$$\begin{array}{rcr} N &\rightarrow ^ { i } & M \\ { {} _ {f} } \downarrow &{} &\swarrow _ {g} \\ E &{} &{} \\ \end{array}$$

Here and below all $R$- modules are supposed to be right $R$- modules. The following conditions on an $R$- module $E$ are equivalent to injectivity: 1) for any exact sequence

$$0 \rightarrow N \rightarrow M \rightarrow L \rightarrow 0$$

the induced sequence

$$0 \rightarrow \mathop{\rm Hom} _ {R} ( N , E ) \rightarrow \mathop{\rm Hom} _ {R} ( M , E ) \rightarrow \mathop{\rm Hom} _ {R} ( L , E ) \rightarrow 0$$

is exact; 2) any exact sequence of $R$- modules of the form

$$0 \rightarrow E \mathop \rightarrow \limits ^ \alpha M \mathop \rightarrow \limits ^ \beta L \rightarrow 0$$

splits, i.e. the submodule $\mathop{\rm Im} \alpha = \mathop{\rm Ker} \beta$ is a direct summand of $M$; 3) $\mathop{\rm Ext} _ {R} ^ {1} ( C , E ) = 0$ for all $R$- modules $C$; and 4) for any right ideal $I$ of $R$ a homomorphism of $R$- modules $f : I \rightarrow E$ can be extended to a homomorphism of $R$- modules $g : R \rightarrow E$( Baer's criterion). There are "enough" injective objects in the category of $R$- modules: Each $R$- module $M$ can be imbedded in an injective module. Moreover, each module $M$ has an injective envelope $E ( M)$, i.e. an injective module containing $M$ in such a way that each non-zero submodule of $E ( M)$ has non-empty intersection with $M$. Any imbedding of a module $M$ into an injective module $E$ can be extended to an imbedding of $E ( M)$ into $E$. Every $R$- module $M$ has an injective resolution

$$0 \rightarrow M \rightarrow E _ {0} \rightarrow E _ {1} \rightarrow \dots ,$$

i.e. an exact sequence of modules in which each module $E _ {i}$, $i \geq 0$, is injective. The length of the shortest injective resolution is called the injective dimension of the module (cf. also Homological dimension).

A direct product of injective modules is an injective module. An injective module $E$ is equal to $E r$ for any $r \in R$ that is not a left zero divisor in $R$, i.e. an injective module is divisible. In particular, an Abelian group is an injective module over the ring $\mathbf Z$ if and only if it is divisible. Let $R$ be a commutative Noetherian ring. Then any injective module over it is a direct sum of injective hulls of modules of the form $R / P$, where $P$ is a prime ideal in $R$.

Injective modules are extensively used in the description of various classes of rings (cf. Homological classification of rings). Thus, all modules over a ring are injective if and only if the ring is semi-simple. The following conditions are equivalent: $R$ is a right Noetherian ring; any direct sum of injective $R$- modules is injective; any injective $R$- module is decomposable as a direct sum of indecomposable $R$- modules. A ring $R$ is right Artinian if and only if every injective module is a direct sum of injective hulls of simple modules. A ring $R$ is right hereditary if and only if all its quotient modules by injective $R$- modules are injective, and also if and only if the sum of two injective submodules of an arbitrary $R$- module is injective. If the ring $R$ is right hereditary and right Noetherian, then every $R$- module contains a largest injective submodule. The projectivity (injectivity) of all injective (projective) $R$- modules is equivalent to $R$ being a quasi-Frobenius ring.

The injective hull of the module $R _ {R}$ plays an important role in the theory of rings of fractions. E.g., if the right singular ideal of a ring $R$ vanishes, if $E$ is the injective hull of the module $R _ {R}$, and if $\Lambda = \mathop{\rm Hom} _ {R} ( E , E )$ is its endomorphism ring, then the $R$- modules $\Lambda _ {R}$ and $E _ {R}$ are isomorphic, $E$ is a ring isomorphic to $\Lambda$ and is also the maximal right ring of fractions of $R$, and $\Lambda \cong E$ is a self-injective right regular ring (in the sense of von Neumann).

In connection with various problems on extending module homomorphisms, some classes of modules $M$ close to injective modules have been considered: quasi-injective modules (if $0 \rightarrow N \rightarrow M$ and $f : N \rightarrow M$, then $f$ can be extended to an endomorphism of $M$); pseudo-injective modules (if $0 \rightarrow N \rightarrow M$ and $f : N \rightarrow M$ is a monomorphism, then $f$ can be extended to an endomorphism of $M$); and small-injective modules (all endomorphisms of submodules can be extended to endomorphisms of $M$). The quasi-injectivity of a module $M$ is equivalent to the invariance of $M$ in its injective hull under endomorphisms of the latter.

#### References

 [1] H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956) [2] S. MacLane, "Homology" , Springer (1963) [3] C. Faith, "Lectures on injective modules and quotient rings" , Springer (1967) [4] D.W. Sharpe, P. Vamos, "Injective modules" , Cambridge Univ. Press (1972)

A ring is called right hereditary if every right ideal is projective or, equivalently, if its right global dimension is $\leq 1$. It is called semi right hereditary if every finitely-generated right ideal is projective. Commutative hereditary integral domains are Dedekind rings; a commutative semi-hereditary integral domain is called a Prüfer ring. A right hereditary ring need not be also left hereditary.