Extension of a module
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
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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
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corresponding to the group . The groups
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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 injective 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) |
Extension of a module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Extension_of_a_module&oldid=17239