Editing Injective module (section)

=== First examples ===

Trivially, the zero module {0} is injective.

Given a [[field (mathematics)|field]] ''k'', every ''k''-[[vector space]] ''Q'' is an injective ''k''-module. Reason: if ''Q'' is a subspace of ''V'', we can find a [[basis of a vector space|basis]] of ''Q'' and extend it to a basis of ''V''. The new extending basis vectors [[linear span|span]] a subspace ''K'' of ''V'' and ''V'' is the internal direct sum of ''Q'' and ''K''. Note that the direct complement ''K'' of ''Q'' is not uniquely determined by ''Q'', and likewise the extending map ''h'' in the above definition is typically not unique.

The rationals '''Q''' (with addition) form an injective abelian group (i.e. an injective '''Z'''-module). The [[factor group]] '''Q'''/'''Z''' and the [[circle group]] are also injective '''Z'''-modules. The factor group '''Z'''/''n'''''Z''' for ''n'' > 1 is injective as a '''Z'''/''n'''''Z'''-module, but ''not'' injective as an abelian group.