Editing Injective module (section)

===Change of rings===
It is important to be able to consider modules over [[subring]]s or [[quotient ring]]s, especially for instance [[polynomial ring]]s. In general, this is difficult, but a number of results are known, {{harv|Lam|1999|p=62}}.

Let ''S'' and ''R'' be rings, and ''P'' be a left-''R'', right-''S'' [[bimodule]] that is [[flat module|flat]] as a left-''R'' module. For any injective right ''S''-module ''M'', the set of [[module homomorphism]]s Hom<sub>''S''</sub>( ''P'', ''M'' ) is an injective right ''R''-module. The same statement holds of course after interchanging left- and right- attributes.

For instance, if ''R'' is a subring of ''S'' such that ''S'' is a flat ''R''-module, then every injective ''S''-module is an injective ''R''-module. In particular, if ''R'' is an integral domain and ''S'' its [[field of fractions]], then every vector space over ''S'' is an injective ''R''-module. Similarly, every injective ''R''[''x'']-module is an injective ''R''-module.

In the opposite direction, a ring homomorphism <math>f: S\to R</math> makes ''R'' into a left-''R'', right-''S'' bimodule, by left and right multiplication. Being [[free module|free]] over itself ''R'' is also [[flat module#Free and projective modules|flat]] as a left ''R''-module. Specializing the above statement for ''P = R'', it says that when ''M'' is an injective right ''S''-module the [[coinduced module]] <math> f_* M = \mathrm{Hom}_S(R, M)</math> is an injective right ''R''-module. Thus, coinduction over ''f'' produces injective ''R''-modules from injective ''S''-modules.

For quotient rings ''R''/''I'', the change of rings is also very clear. An ''R''-module is an ''R''/''I''-module precisely when it is annihilated by ''I''. The submodule ann<sub>''I''</sub>(''M'') = { ''m'' in ''M'' : ''im'' = 0 for all ''i'' in ''I'' } is a left submodule of the left ''R''-module ''M'', and is the largest submodule of ''M'' that is an ''R''/''I''-module. If ''M'' is an injective left ''R''-module, then ann<sub>''I''</sub>(''M'') is an injective left ''R''/''I''-module. Applying this to ''R''='''Z''', ''I''=''n'''''Z''' and ''M''='''Q'''/'''Z''', one gets the familiar fact that '''Z'''/''n'''''Z''' is injective as a module over itself. While it is easy to convert injective ''R''-modules into injective ''R''/''I''-modules, this process does not convert injective ''R''-resolutions into injective ''R''/''I''-resolutions, and the homology of the resulting complex is one of the early and fundamental areas of study of relative homological algebra.

The textbook {{harv|Rotman|1979|p=103}} has an erroneous proof that [[localization of a ring|localization]] preserves injectives, but a counterexample was given in {{harv|Dade|1981}}.