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Difference between revisions of "Extension of a module"

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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 injective 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 cover.
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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 cover]].
  
 
====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>

Revision as of 20:26, 30 October 2016

Any module containing the given module as a submodule. Usually one fixes a quotient module , that is, an extension of the module by the module is an exact sequence

Such a module always exists (for example, the direct sum of and ), but need not be uniquely determined by and . Both in the theory of modules and in its applications there is a need to describe all different extensions of a module by a module . To this end one defines an equivalence relation on the class of all extensions of by as well as a binary operation (called Baer multiplication) on the set of equivalence classes, which thus becomes an Abelian group , where is the ring over which is a module. This construction can be extended to -fold extensions of by , i.e. to exact sequences of the form

corresponding to the group . The groups

are the derived functors of the functor , and may be computed using a projective resolution of or an injective resolution of . An extension of is called essential if is the only submodule of with . 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 is called the injective hull or envelope of . The notion can be defined in any Abelian category, cf. [a1]. The dual notion is that of a projective cover.

References

[a1] C. Faith, "Algebra: rings, modules, and categories" , 1 , Springer (1973)
How to Cite This Entry:
Extension of a module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Extension_of_a_module&oldid=17239
This article was adapted from an original article by V.E. Govorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article