# 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 functors 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.

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. [a1]. The dual notion is that of a projective covering.