View source for Extension of a module

Any module $X$ containing the given module $A$ as a submodule. Usually one fixes a quotient module $X/A$, that is, an extension of the module $A$ by the module $B$ is an [[exact sequence]]
$$
0 \rightarrow A \rightarrow X \rightarrow B \rightarrow 0 \ .
$$

Such a module $X$ always exists: for example, the [[direct sum]] of $A$ and $B$ always forms the ''split extension''; but $X$ need not be uniquely determined by $A$ and $B$. Both in the theory of modules and in its applications there is a need to describe all different extensions of a module $A$ by a module $B$. To this end one defines an equivalence relation on the class of all extensions of $A$ by $B$ as well as a binary operation (called [[Baer multiplication]]) on the set of equivalence classes, which thus becomes an Abelian group $\mathrm{Ext}^1_R(A,B)$, where $R$ is the ring over which $A$ is a module. This construction can be extended to $n$-fold extensions of $A$ by $B$, i.e. to exact sequences of the form
$$
0 \rightarrow A \rightarrow X_{n-1} \rightarrow \cdots \rightarrow X_0 \rightarrow B \rightarrow 0
$$
corresponding to the group $\mathrm{Ext}^n_R(A,B)$. The groups $\mathrm{Ext}^n_R(A,B)$, $n=1,2,\ldots$, are the [[derived functor]]s of the functor $\mathrm{Hom}_R(A,B)$, and may be computed using a [[projective resolution]] of $A$ or an [[injective resolution]] of $B$. An extension $X$ of $A$ is called ''essential'' if $S = 0$ is the only submodule of $X$ with $S \cap A = 0$ (that is, $A$ is an [[essential submodule]] of $X$). Every module has a maximal essential extension and this is the minimal [[injective module]] containing the given one.

For references see [[Extension of a group]].



====Comments====
The minimal injective module containing $A$ is called the [[injective hull]] or envelope of $A$. The notion can be defined in any Abelian category, cf. [[#References|[a1]]]. The dual notion is that of a [[projective covering]].

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Faith,   "Algebra: rings, modules, and categories" , '''1''' , Springer  (1973)</TD></TR>
</table>

{{TEX|done}}