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− | Any module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037000/e0370001.png" /> containing the given module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037000/e0370002.png" /> as a submodule. Usually one fixes a quotient module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037000/e0370003.png" />, that is, an extension of the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037000/e0370004.png" /> by the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037000/e0370005.png" /> is an [[Exact sequence|exact sequence]] | + | 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]] |
| + | $$ |
| + | 0 \rightarrow A \rightarrow X \rightarrow B \rightarrow 0 \ . |
| + | $$ |
| | | |
− | <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/e/e037/e037000/e0370006.png" /></td> </tr></table>
| + | 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. |
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− | Such a module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037000/e0370007.png" /> always exists (for example, the direct sum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037000/e0370008.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037000/e0370009.png" />), but need not be uniquely determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037000/e03700010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037000/e03700011.png" />. Both in the theory of modules and in its applications there is a need to describe all different extensions of a module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037000/e03700012.png" /> by a module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037000/e03700013.png" />. To this end one defines an equivalence relation on the class of all extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037000/e03700014.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037000/e03700015.png" /> as well as a binary operation (called [[Baer multiplication|Baer multiplication]]) on the set of equivalence classes, which thus becomes an Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037000/e03700016.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037000/e03700017.png" /> is the ring over which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037000/e03700018.png" /> is a module. This construction can be extended to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037000/e03700020.png" />-fold extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037000/e03700021.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037000/e03700022.png" />, i.e. to exact sequences of the form
| + | For references see [[Extension of a group]]. |
− | | |
− | <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/e/e037/e037000/e03700023.png" /></td> </tr></table>
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− | corresponding to the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037000/e03700024.png" />. The groups
| |
− | | |
− | <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/e/e037/e037000/e03700025.png" /></td> </tr></table>
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− | | |
− | are the derived functors of the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037000/e03700026.png" />, and may be computed using a projective resolution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037000/e03700027.png" /> or an injective resolution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037000/e03700028.png" />. An extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037000/e03700029.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037000/e03700030.png" /> is called essential if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037000/e03700031.png" /> is the only submodule of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037000/e03700032.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037000/e03700033.png" />. Every module has a maximal essential extension and this is the minimal [[Injective module|injective module]] containing the given one.
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− | For references see [[Extension of a group|Extension of a group]]. | |
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| ====Comments==== | | ====Comments==== |
− | The minimal injective module containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037000/e03700034.png" /> is called the [[injective hull]] or envelope of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037000/e03700035.png" />. The notion can be defined in any Abelian category, cf. [[#References|[a1]]]. The dual notion is that of a [[projective covering]]. | + | 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==== | | ====References==== |
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> C. Faith, "Algebra: rings, modules, and categories" , '''1''' , Springer (1973)</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> |
| + | </table> |
| + | |
| + | {{TEX|done}} |
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.
References
[a1] | C. Faith, "Algebra: rings, modules, and categories" , 1 , Springer (1973) |