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An [[Injective object|injective object]] in the category of (right) modules over an associative [[ring with identity]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i0512101.png" />, i.e. an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i0512102.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i0512103.png" /> such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i0512104.png" />-modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i0512105.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i0512106.png" />, for any monomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i0512107.png" />, and for any homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i0512108.png" /> there is a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i0512109.png" /> that makes the following diagram commutative
+
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121010.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
Here and below all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121011.png" />-modules are supposed to be right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121012.png" />-modules. The following conditions on an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121013.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121014.png" /> are equivalent to injectivity: 1) for any [[Exact sequence|exact sequence]]
+
An [[Injective object|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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121015.png" /></td> </tr></table>
+
$$
 +
 
 +
\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|exact sequence]]
 +
 
 +
$$
 +
0  \rightarrow  N  \rightarrow  M  \rightarrow  L  \rightarrow  0
 +
$$
  
 
the induced sequence
 
the induced sequence
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121016.png" /></td> </tr></table>
+
$$
 +
0 \rightarrow  \mathop{\rm Hom} _ {R} ( N , E )  \rightarrow  \mathop{\rm Hom} _ {R} ( M , E )  \rightarrow  \mathop{\rm Hom} _ {R} ( L , E )  \rightarrow  0
 +
$$
  
is exact; 2) any exact sequence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121017.png" />-modules of the form
+
is exact; 2) any exact sequence of $  R $-
 +
modules of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121018.png" /></td> </tr></table>
+
$$
 +
0 \rightarrow  E  \mathop \rightarrow \limits ^  \alpha    M  \mathop \rightarrow \limits ^  \beta    L  \rightarrow  0
 +
$$
  
splits, i.e. the submodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121019.png" /> is a direct summand of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121020.png" />; 3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121021.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121022.png" />-modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121023.png" />; and 4) for any right ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121024.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121025.png" /> a homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121026.png" />-modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121027.png" /> can be extended to a homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121028.png" />-modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121029.png" /> (Baer's criterion). There are  "enough"  injective objects in the category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121030.png" />-modules: Each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121031.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121032.png" /> can be imbedded in an injective module. Moreover, each module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121033.png" /> has an [[injective evelope]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121034.png" />, i.e. an injective module containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121035.png" /> in such a way that each non-zero submodule of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121036.png" /> has non-empty intersection with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121037.png" />. Any imbedding of a module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121038.png" /> into an injective module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121039.png" /> can be extended to an imbedding of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121040.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121041.png" />. Every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121042.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121043.png" /> has an injective resolution
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121044.png" /></td> </tr></table>
+
$$
 +
0 \rightarrow  M  \rightarrow  E _ {0}  \rightarrow  E _ {1}  \rightarrow \dots ,
 +
$$
  
i.e. an exact sequence of modules in which each module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121046.png" />, is injective. The length of the shortest injective resolution is called the injective dimension of the module (cf. also [[Homological dimension|Homological dimension]]).
+
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|Homological dimension]]).
  
A direct product of injective modules is an injective module. An injective module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121047.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121048.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121049.png" /> that is not a left zero divisor in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121050.png" />, i.e. an injective module is divisible. In particular, an Abelian group is an injective module over the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121051.png" /> if and only if it is divisible. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121052.png" /> be a commutative Noetherian ring. Then any injective module over it is a direct sum of injective hulls of modules of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121053.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121054.png" /> is a prime ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121055.png" />.
+
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|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: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121056.png" /> is a right Noetherian ring; any direct sum of injective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121057.png" />-modules is injective; any injective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121058.png" />-module is decomposable as a direct sum of indecomposable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121059.png" />-modules. A ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121060.png" /> is right Artinian if and only if every injective module is a direct sum of injective hulls of simple modules. A ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121061.png" /> is right hereditary if and only if all its quotient modules by injective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121062.png" />-modules are injective, and also if and only if the sum of two injective submodules of an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121063.png" />-module is injective. If the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121064.png" /> is right hereditary and right Noetherian, then every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121065.png" />-module contains a largest injective submodule. The projectivity (injectivity) of all injective (projective) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121066.png" />-modules is equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121067.png" /> being a [[Quasi-Frobenius ring|quasi-Frobenius ring]].
+
Injective modules are extensively used in the description of various classes of rings (cf. [[Homological classification of rings|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|quasi-Frobenius ring]].
  
The injective hull of the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121068.png" /> plays an important role in the theory of rings of fractions. E.g., if the right singular ideal of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121069.png" /> vanishes, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121070.png" /> is the injective hull of the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121071.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121072.png" /> is its endomorphism ring, then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121073.png" />-modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121074.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121075.png" /> are isomorphic, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121076.png" /> is a ring isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121077.png" /> and is also the maximal right ring of fractions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121078.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121079.png" /> is a self-injective right [[Regular ring (in the sense of von Neumann)|regular ring (in the sense of von Neumann)]].
+
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)|regular ring (in the sense of von Neumann)]].
  
In connection with various problems on extending module homomorphisms, some classes of modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121080.png" /> close to injective modules have been considered: quasi-injective modules (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121081.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121082.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121083.png" /> can be extended to an endomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121084.png" />); pseudo-injective modules (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121085.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121086.png" /> is a monomorphism, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121087.png" /> can be extended to an endomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121088.png" />); and small-injective modules (all endomorphisms of submodules can be extended to endomorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121089.png" />). The quasi-injectivity of a module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121090.png" /> is equivalent to the invariance of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121091.png" /> in its injective hull under endomorphisms of the latter.
+
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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Cartan,  S. Eilenberg,  "Homological algebra" , Princeton Univ. Press  (1956)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. MacLane,  "Homology" , Springer  (1963)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C. Faith,  "Lectures on injective modules and quotient rings" , Springer  (1967)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  D.W. Sharpe,  P. Vamos,  "Injective modules" , Cambridge Univ. Press  (1972)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Cartan,  S. Eilenberg,  "Homological algebra" , Princeton Univ. Press  (1956)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. MacLane,  "Homology" , Springer  (1963)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C. Faith,  "Lectures on injective modules and quotient rings" , Springer  (1967)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  D.W. Sharpe,  P. Vamos,  "Injective modules" , Cambridge Univ. Press  (1972)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
A ring is called right hereditary if every right ideal is projective or, equivalently, if its right global dimension is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051210/i05121092.png" />. 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.
+
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====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Faith,  "Algebra: rings, modules, and categories" , '''1''' , Springer  (1973)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.C. McConnell,  J.C. Robson,  "Noncommutative Noetherian rings" , Wiley  (1987)  pp. Part I, Chapt. 2</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Faith,  "Algebra: rings, modules, and categories" , '''1''' , Springer  (1973)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.C. McConnell,  J.C. Robson,  "Noncommutative Noetherian rings" , Wiley  (1987)  pp. Part I, Chapt. 2</TD></TR></table>

Latest revision as of 22:12, 5 June 2020


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
How to Cite This Entry:
Injective module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Injective_module&oldid=39556
This article was adapted from an original article by A.V. MikhalevA.A. Tuganbaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article