TheoremStatement: Literature-sourcedProof: Literature-sourcedprecheck passaudited 2026-08-13

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Equivalent characterizations of injective modules

Statement

For a left $R$ -module $I$ , the following are equivalent:

$I$ is injective;
every short exact sequence $0 \to I \to E \to C \to 0$ splits;
${Hom}_{R} (-, I)$ takes every short exact sequence to a short exact sequence.

These equivalences use no choice principle.

Facts & Assumptions

Given: A left $R$ -module $I$ .

[F1]

Injectivity is the extension property along every module monomorphism (Injective modules and the extension property).

[L1]

A short exact sequence splits exactly when its monomorphism has a retraction (The splitting lemma for short exact sequences of modules).

[L2]

Applying ${Hom}_{R} (-, I)$ to an exact sequence $A \to B \to C \to 0$ gives an exact sequence $0 \to {Hom}_{R} (C, I) \to {Hom}_{R} (B, I) \to {Hom}_{R} (A, I)$ (Covariant and contravariant $Hom$ are left exact).

[F2]

Quotient modules have the usual coset operations (Quotient module $M / N$ with scalar multiplication on additive cosets).

Proof

technique · direct

1.1

If $I$ is injective and $0 \to I \overset{j}{\to} E \to C \to 0$ is short exact, extend ${id}_{I}$ along $j$ using [F1]. The extension is a retraction, so [L1] makes the sequence split.

assume-hypF1L1

1.2

Conversely, assume every short exact sequence beginning in $I$ splits. Given a monomorphism $u : A \to B$ and $f : A \to I$ , let $S = {(f (a), - u (a)) : a \in A} \leq I \oplus B$ and $P = (I \oplus B) / S$ .

assume-hypF2construct

1.3

If $I$ is injective, every map $A \to I$ extends across the monomorphism in a short exact sequence $0 \to A \to B \to C \to 0$ , so the final precomposition map in [L2] is surjective; hence ${Hom}_{R} (-, I)$ is exact.

assume-hypF1L2

1.4

Conversely, if ${Hom}_{R} (-, I)$ takes short exact sequences to short exact sequences, apply it to $0 \to A \overset{u}{\to} B \to B / u (A) \to 0$ . Surjectivity of $u^{*}$ extends every $A \to I$ across $u$ , so [F1] makes $I$ injective.

assume-hypF1F2L2

2.1

The map $j : I \to P$ , $j (t) = [(t, 0)]$ , is injective: if $(t, 0) = (f (a), - u (a))$ , injectivity of $u$ gives $a = 0$ and $t = 0$ . Hence $0 \to I \overset{j}{\to} P \to P / j (I) \to 0$ is short exact and splits by hypothesis; let $r : P \to I$ retract $j$ .

step 1.2L1F2

3.1

Define $\tilde{f} : B \to I$ by $\tilde{f} (b) = r ([(0, b)])$ . In $P$ , $[(0, u (a))] = [(f (a), 0)]$ , so $\tilde{f} (u (a)) = r (j (f (a))) = f (a)$ . Thus $I$ is injective by [F1].

step 1.2step 2.1F1construct

4.1

Steps 1.1, 1.2, 2.1, and 3.1 prove $1 \Leftrightarrow 2$ , while steps 1.3 and 1.4 prove $1 \Leftrightarrow 3$ .

step 1.1step 1.2step 2.1step 3.1step 1.3step 1.4∎

Depends on

Used by

Over a finite group algebra in defining characteristic, finite-dimensional projective and injective modules coincide Corollary
Every short exact sequence of modules splits False statement
Semisimple rings have vanishing positive Tor and Ext Proposition

Dependency tree · two levels

15 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

A. Kleshchev, Lectures on Abstract Algebra for Graduate Students, sections 3.6, 3.14, and 3.15 (standard reference, not scraped)
The Stacks Project, Algebra (standard reference, not scraped)
P. Hekmati, Homological Algebra, section 3.1 (standard reference, not scraped)