In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers. Specifically, if Q is a submodule of some other module, then it is already a direct summand of that module; also, given a submodule of a module Y, any module homomorphism from this submodule to Q can be extended to a homomorphism from all of Y to Q. This concept is dual to that of projective modules. Injective modules were introduced in (Baer 1940) and are discussed in some detail in the textbook (Lam 1999, §3).

Injective modules have been heavily studied, and a variety of additional notions are defined in terms of them: Injective cogenerators are injective modules that faithfully represent the entire category of modules. Injective resolutions measure how far from injective a module is in terms of the injective dimension and represent modules in the derived category. Injective hulls are maximal essential extensions, and turn out to be minimal injective extensions. Over a Noetherian ring, every injective module is uniquely a direct sum of indecomposable modules, and their structure is well understood. An injective module over one ring may be not injective over another, but there are well-understood methods of changing rings which handle special cases. Rings which are themselves injective modules have a number of interesting properties and include rings such as group rings of finite groups over fields. Injective modules include divisible groups and are generalized by the notion of injective objects in category theory.

A left module ${\displaystyle Q}$ over the ring ${\displaystyle R}$ is injective if it satisfies one (and therefore all) of the following equivalent conditions:

Injective right ${\displaystyle R}$-modules are defined analogously.

Trivially, the zero module ${\displaystyle \{0\}}$ is injective.

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

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

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 summands are the localizations ${\displaystyle R_{\mathfrak {p}}/R}$ for the nonzero prime ideals ${\displaystyle {\mathfrak {p}}}$. 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.