Editing Injective module (section)

===Injective resolutions===
Every module ''M'' also has an injective [[resolution (algebra)|resolution]]: an [[exact sequence]] of the form
:0 → ''M'' → ''I''<sup>0</sup> → ''I''<sup>1</sup> → ''I''<sup>2</sup> → ...
where the ''I''<sup> ''j''</sup> are injective modules. Injective resolutions can be used to define [[derived functor]]s such as the [[Ext functor]].

The ''length'' of a finite injective resolution is the first index ''n'' such that ''I''<sup>''n''</sup> is nonzero and ''I''<sup>''i''</sup>&nbsp;=&nbsp;0 for ''i'' greater than ''n''. If a module ''M'' admits a finite injective resolution, the minimal length among all finite injective resolutions of ''M'' is called its injective dimension and denoted id(''M''). If ''M'' does not admit a finite injective resolution, then by convention the injective dimension is said to be infinite. {{harv|Lam|1999|loc=§5C}} As an example, consider a module ''M'' such that id(''M'')&nbsp;=&nbsp;0. In this situation, the exactness of the sequence 0 → ''M'' → ''I''<sup>0</sup> → 0 indicates that the arrow in the center is an isomorphism, and hence ''M'' itself is injective.<ref>A module isomorphic to an injective module is of course injective.</ref>

Equivalently, the injective dimension of ''M'' is the minimal integer (if there is such, otherwise ∞) ''n'' such that Ext{{su|p=''N''|b=''A''}}(–,''M'') = 0 for all ''N'' > ''n''.