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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)

Comments

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.

References

[a1]	C. Faith, "Algebra: rings, modules, and categories" , 1 , Springer (1973)
[a2]	J.C. McConnell, J.C. Robson, "Noncommutative Noetherian rings" , Wiley (1987) pp. Part I, Chapt. 2