# Injective module

An injective module is a module J over a ring R with the extension property that every homomorphism from a submodule A of any R-module B into J extends to a homomorphism from all of B into J. Equivalently, the functor Hom_R(−, J) is exact, so it turns short exact sequences into short exact sequences. Injectivity is the categorical dual of projectivity: where a projective module receives maps from free modules, an injective module absorbs maps into itself.<sup>[1](https://math.hodapp.com/lectures/higher/homological-algebra/derived-functors.html)</sup>

| Key fact | Detail |
|---|---|
| Defining property | For every injection A → B and map A → J, the map extends to B → J; equivalently Hom_R(−, J) is exact<sup>[1](https://math.hodapp.com/lectures/higher/homological-algebra/derived-functors.html)</sup> |
| Baer's criterion | J is injective iff every homomorphism from a left ideal I of R into J extends to a homomorphism R → J<sup>[2](https://www.math.buffalo.edu/~badzioch/MTH620/Lecture_Notes_files/algebra619-620.pdf)</sup> |
| Enough injectives | Every R-module M embeds in an injective R-module, so the category R-Mod has enough injective objects<sup>[3](https://androma.org/theorems/4560)</sup> |
| Injective envelope | Every module M has an injective envelope E(M), a maximal essential extension of M<sup>[4](https://therisingsea.org/notes/mast90068/lecture21.pdf)</sup> |
| Over a PID | Injective R-modules are exactly the divisible modules; Q is an injective Z-module<sup>[2](https://www.math.buffalo.edu/~badzioch/MTH620/Lecture_Notes_files/algebra619-620.pdf)</sup> |
| Matlis classification (1958) | Over a commutative Noetherian ring, indecomposable injectives correspond bijectively to prime ideals, P ↔ E(R/P)<sup>[5](https://msp.org/pjm/1958/8-3/pjm-v8-n3-p13-p.pdf)</sup> |
| Bass–Papp theorem | Direct sums of injective modules are injective if and only if the ring is Noetherian<sup>[6](https://www.sciencedirect.com/topics/mathematics/injective-module)</sup> |

## Definition and Baer's criterion

The extension property is the formal definition: given any monomorphism of R-modules ι: A → B and any homomorphism f: A → J, there exists a homomorphism f̄: B → J with f̄ι = f. An equivalent splitting formulation says that if J is a submodule of M, then J is a direct summand of M.<sup>[1](https://math.hodapp.com/lectures/higher/homological-algebra/derived-functors.html)</sup><sup> • </sup><sup>[2](https://www.math.buffalo.edu/~badzioch/MTH620/Lecture_Notes_files/algebra619-620.pdf)</sup>

Checking this property for all pairs of modules would be impractical. Baer's criterion reduces the test to the ring itself: for a ring R with identity, an R-module J is injective if and only if, for every left ideal I of R and every homomorphism f: I → J, there is a homomorphism f̄: R → J extending f.<sup>[2](https://www.math.buffalo.edu/~badzioch/MTH620/Lecture_Notes_files/algebra619-620.pdf)</sup> The criterion appeared in [Reinhold Baer](https://www.edgepedia.org/reinhold-baer)'s 1940 paper *Abelian groups that are direct summands of every containing abelian group*, published in the Bulletin of the American Mathematical Society, which introduced injectivity for modules as a generalization of the divisibility of abelian groups.<sup>[7](https://mathworld.wolfram.com/BaersCriterion.html)</sup><sup> • </sup><sup>[8](https://theses.gla.ac.uk/76508/)</sup>

## Existence of injectives and the injective envelope

Unlike free modules, which surject onto every module by construction, injective modules are not present in every category automatically. For modules over a ring with identity, existence holds: for any R-module M there is a monomorphism j: M → J with J injective, so the category of R-modules has enough injectives.<sup>[2](https://www.math.buffalo.edu/~badzioch/MTH620/Lecture_Notes_files/algebra619-620.pdf)</sup><sup> • </sup><sup>[3](https://androma.org/theorems/4560)</sup> The standard proof embeds M into its double character module with respect to Q/Z, using the facts that Q/Z is an injective cogenerator of abelian groups and that Hom_Z(N, Q/Z) is injective as a left R-module for every right R-module N.<sup>[3](https://androma.org/theorems/4560)</sup>

Among the injective modules containing a given M there is a smallest one in an essential sense. An essential extension of M is a module E containing M such that every nonzero submodule of E meets M; Baer discovered that a maximal essential extension exists, and Eckmann and Schöpf gave the characterization of this minimal injective extension, now called the injective hull or injective envelope E(M).<sup>[6](https://www.sciencedirect.com/topics/mathematics/injective-module)</sup> The construction uses [Zorn's lemma](https://www.edgepedia.org/zorns-lemma) on the poset of essential extensions of M inside a fixed injective module, and the maximal element is itself injective because it admits no proper essential extension.<sup>[4](https://therisingsea.org/notes/mast90068/lecture21.pdf)</sup> The injective hull is the dual of the [projective cover](https://www.edgepedia.org/projective-cover), and it plays a significant role in Noetherian commutative algebra and in algebraic topology.<sup>[6](https://www.sciencedirect.com/topics/mathematics/injective-module)</sup>

## Examples

The basic example is Q as a module over Z. Given an ideal I of Z and a homomorphism f: I → Q, the relation rf(s) = sf(r) forces f(r)/r to be the same rational number a for every nonzero r in I, and the map r ↦ ra extends f to all of R; by Baer's criterion, Q is injective.<sup>[2](https://www.math.buffalo.edu/~badzioch/MTH620/Lecture_Notes_files/algebra619-620.pdf)</sup> More generally, for an integral domain R with field of fractions K, the module K is injective.<sup>[2](https://www.math.buffalo.edu/~badzioch/MTH620/Lecture_Notes_files/algebra619-620.pdf)</sup>

**Divisibility.** Over a principal ideal domain, injective modules are exactly the divisible modules, those in which multiplication by every nonzero ring element is surjective. Since Z is a PID, injective Z-modules are precisely the divisible abelian groups, and a divisible abelian group decomposes as a direct sum of copies of Q and of Prüfer p-groups Z(p^∞) for various primes p.<sup>[2](https://www.math.buffalo.edu/~badzioch/MTH620/Lecture_Notes_files/algebra619-620.pdf)</sup> Over a field, every module is both projective and injective, since every short exact sequence of vector spaces splits.<sup>[1](https://math.hodapp.com/lectures/higher/homological-algebra/derived-functors.html)</sup>

Quotients behave well in this setting: over a PID, a quotient J/K of an injective module J is again injective, while over a general ring a quotient of an injective module need not be injective.<sup>[2](https://www.math.buffalo.edu/~badzioch/MTH620/Lecture_Notes_files/algebra619-620.pdf)</sup>

## Structure over Noetherian rings

The decisive structure theorem is due to Eben Matlis, a mathematician whose 1958 paper *Injective modules over Noetherian rings* in the Pacific Journal of Mathematics studied the structure and properties of injective modules over Noetherian rings.<sup>[5](https://msp.org/pjm/1958/8-3/pjm-v8-n3-p13-p.pdf)</sup> Two statements together describe all injectives over a commutative Noetherian ring R:<sup>[9](https://stacks.math.columbia.edu/tag/08YA)</sup>

1. Every injective R-module is a direct sum of indecomposable injective modules, and the decomposition is unique up to isomorphism.<sup>[9](https://stacks.math.columbia.edu/tag/08YA)</sup><sup> • </sup><sup>[5](https://msp.org/pjm/1958/8-3/pjm-v8-n3-p13-p.pdf)</sup>
2. The indecomposable injectives are exactly the injective hulls E(R/P) as P runs over the prime ideals of R, giving a bijective correspondence between prime ideals and isomorphism classes of indecomposable injectives.<sup>[5](https://msp.org/pjm/1958/8-3/pjm-v8-n3-p13-p.pdf)</sup><sup> • </sup><sup>[10](https://londmathsoc.onlinelibrary.wiley.com/doi/full/10.1112/blms.12689)</sup>

When P = 0, so that R is an integral domain, E(R/P) is the quotient field of R; for nonzero P the module generalizes the Prüfer p-group Z(p^∞).<sup>[5](https://msp.org/pjm/1958/8-3/pjm-v8-n3-p13-p.pdf)</sup> The endomorphism ring of E(R/P) is isomorphic to the completion of the localization R_P, and Matlis's 1958 paper also established a duality between Noetherian modules and Artinian modules over a complete local ring, with the duality functor Hom_R(−, E(R/m)) for the maximal ideal m.<sup>[10](https://londmathsoc.onlinelibrary.wiley.com/doi/full/10.1112/blms.12689)</sup> Matlis also connected this decomposition theory with Lasker–Noether primary decomposition.<sup>[6](https://www.sciencedirect.com/topics/mathematics/injective-module)</sup>

**The Bass–Papp theorem** marks where Noetherianness enters. [Hyman Bass](https://www.edgepedia.org/hyman-bass) and Goro Azumaya's circle of results established in 1956 that every direct sum of injective R-modules is injective if and only if R is Noetherian; over a non-[Noetherian ring](https://www.edgepedia.org/noetherian-ring), some direct sum of injectives fails to be injective.<sup>[6](https://www.sciencedirect.com/topics/mathematics/injective-module)</sup> A derived-category refinement due to Neeman and later work characterizes when the category of injectives is closed under arbitrary direct sums in the derived category in terms of Noetherian-type conditions on the cohomology modules.<sup>[11](https://arxiv.org/abs/1709.01479)</sup>

## Injective resolutions and derived functors

An injective resolution of a module M is an exact sequence 0 → M → I⁰ → I¹ → I² → ⋯ in which each Iⁿ is injective. Because the category of modules has enough injectives, every module has an injective resolution.<sup>[1](https://math.hodapp.com/lectures/higher/homological-algebra/derived-functors.html)</sup><sup> • </sup><sup>[6](https://www.sciencedirect.com/topics/mathematics/injective-module)</sup> Injective resolutions are the computational device for right derived functors of left exact functors: applying Hom_R(M, −) to an injective resolution of the second variable and taking cohomology yields Ext^i_R(M, N), and the balancing theorem shows the result agrees with the computation from projective resolutions of the first variable.<sup>[1](https://math.hodapp.com/lectures/higher/homological-algebra/derived-functors.html)</sup>

The same machinery extends beyond modules. The Stacks project shows that the category of sheaves of O-modules on a site has enough injectives, indeed with functorial injective embeddings, which is the foundation for sheaf cohomology.<sup>[12](https://stacks.math.columbia.edu/download/injectives.pdf)</sup>

## History

Baer introduced injectivity in 1940, defining it for modules as a generalization of the divisibility of abelian groups and proving the criterion now bearing his name.<sup>[8](https://theses.gla.ac.uk/76508/)</sup><sup> • </sup><sup>[7](https://mathworld.wolfram.com/BaersCriterion.html)</sup> In the same period the transfinite-induction argument showing that every R-module embeds in an injective one was in place, and the functor Ext of abelian extensions emerged from these constructions.<sup>[13](https://www.e-periodica.ch/cntmng?pid=ens-001%3A1978%3A24%3A%3A9)</sup> Matlis's 1958 classification of indecomposable injectives over commutative Noetherian rings, together with his duality theory, formed an important part of the commutative algebra landscape of the 1960s.<sup>[5](https://msp.org/pjm/1958/8-3/pjm-v8-n3-p13-p.pdf)</sup><sup> • </sup><sup>[10](https://londmathsoc.onlinelibrary.wiley.com/doi/full/10.1112/blms.12689)</sup>

## References

1. [Derived functors and resolutions, Math foundations lecture notes](https://math.hodapp.com/lectures/higher/homological-algebra/derived-functors.html)
2. [Notes on Algebra, MTH 619/620 lecture notes, University at Buffalo](https://www.math.buffalo.edu/~badzioch/MTH620/Lecture_Notes_files/algebra619-620.pdf)
3. [Enough Injectives in the Category of Modules, Androma Theorem #4560](https://androma.org/theorems/4560)
4. [Injective envelopes, MAST90068 lecture 21 notes, The Rising Sea](https://therisingsea.org/notes/mast90068/lecture21.pdf)
5. [Eben Matlis (1958). Injective modules over Noetherian rings. Pacific Journal of Mathematics 8(3), 511–528.](https://msp.org/pjm/1958/8-3/pjm-v8-n3-p13-p.pdf)
6. [Injective Module, ScienceDirect Topics overview](https://www.sciencedirect.com/topics/mathematics/injective-module)
7. [Baer's Criterion, Wolfram MathWorld](https://mathworld.wolfram.com/BaersCriterion.html)
8. [CS-Modules and Generalizations, PhD thesis, University of Glasgow](https://theses.gla.ac.uk/76508/)
9. [Proposition 47.5.9: Structure of injective modules over Noetherian rings, The Stacks project](https://stacks.math.columbia.edu/tag/08YA)
10. [Peter Vámos, 1940–2020 (obituary by R. Y. Sharp), Bulletin of the London Mathematical Society, 2022](https://londmathsoc.onlinelibrary.wiley.com/doi/full/10.1112/blms.12689)
11. [Injective DG-modules over non-positive DG-rings, arXiv:1709.01479](https://arxiv.org/abs/1709.01479)
12. [Chapter on Injectives, The Stacks Project](https://stacks.math.columbia.edu/download/injectives.pdf)
13. [Historical note on homological algebra, E-periodica](https://www.e-periodica.ch/cntmng?pid=ens-001%3A1978%3A24%3A%3A9)

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