Editing Injective module (section)

=== Commutative examples ===

More generally, for any [[integral domain]] ''R'' with field of fractions ''K'', the ''R''-module ''K'' is an injective ''R''-module, and indeed the smallest injective ''R''-module containing ''R''. For any [[Dedekind domain]], the [[quotient module]] ''K''/''R'' is also injective, and its [[indecomposable module|indecomposable]] summands are the [[localization of a ring|localizations]] <math>R_{\mathfrak{p}}/R</math> for the nonzero [[prime ideal]]s <math>\mathfrak{p}</math>. The [[zero ideal]] is also prime and corresponds to the injective ''K''. In this way there is a 1-1 correspondence between prime ideals and indecomposable injective modules.

A particularly rich theory is available for [[commutative ring|commutative]] [[noetherian ring]]s due to [[Eben Matlis]], {{harv|Lam|1999|loc=§3I}}. Every injective module is uniquely a direct sum of indecomposable injective modules, and the indecomposable injective modules are uniquely identified as the injective hulls of the quotients ''R''/''P'' where ''P'' varies over the [[prime spectrum]] of the ring. The injective hull of ''R''/''P'' as an ''R''-module is canonically an ''R''<sub>''P''</sub> module, and is the ''R''<sub>''P''</sub>-injective hull of ''R''/''P''. In other words, it suffices to consider [[local ring]]s. The [[endomorphism ring]] of the injective hull of ''R''/''P'' is the [[completion (ring theory)|completion]] <math>\hat R_P</math> of ''R'' at ''P''.<ref>{{Cite web|url=https://stacks.math.columbia.edu/tag/08Z6|title=Lemma 47.7.5 (08Z6)—The Stacks project|website=stacks.math.columbia.edu|access-date=2020-02-25}}</ref>

Two examples are the injective hull of the '''Z'''-module '''Z'''/''p'''''Z''' (the [[Prüfer group]]), and the injective hull of the ''k''[''x'']-module ''k'' (the ring of inverse polynomials). The latter is easily described as ''k''[''x'',''x''<sup>−1</sup>]/''xk''[''x'']. This module has a basis consisting of "inverse monomials", that is ''x''<sup>−''n''</sup> for ''n'' = 0, 1, 2, …. Multiplication by scalars is as expected, and multiplication by ''x'' behaves normally except that ''x''·1 = 0. The endomorphism ring is simply the ring of [[formal power series]].