Injective Modules and Dualization

Executive summary

Everything reverses, but the proofs do not

Injectivity is projectivity with every arrow reversed: maps into an injective module extend along injections. The formal duality is exact, but the two notions behave quite differently in practice. Free modules make projectives easy to produce, and there is no dual construction; instead, Baer's criterion reduces injectivity to a test on ideals, and a separate argument shows every module embeds in an injective one — which is what injective resolutions need.

Learning objectives

State the extension property defining injectivity.
Apply Baer's criterion to test a module.
Characterise injective abelian groups as the divisible ones.
Explain why enough injectives requires an argument where projectives do not.
Describe the injective hull and its uniqueness.

Section 01The extension property

I is injective when for every injection A ↪ B and every map A → I, an extension B → I exists.

Projective PLifts along surjections

Every 0 → A → B → P → 0 splits. Hom(P, −) is exact. Produced from free modules.

Injective IExtends along injections

Every 0 → I → B → C → 0 splits. Hom(−, I) is exact. Requires a construction with no free analogue.

The duality is formal, not practical

Reversing arrows turns every statement about projectives into a true statement about injectives. It does not turn the constructions into each other: there is no dual of a basis. Producing injectives is genuinely harder, which is why Baer's criterion matters.

Section 02Baer's criterion

AlgorithmBaer's criterionin: a module I → out: whether I is injective

To test whether I is injective, it suffices to consider injections of ideals into the ring itself.
For every left ideal J ⊆ Λ and every homomorphism f: J → I:
check that f extends to a homomorphism Λ → I. Equivalently, f is given by right multiplication by some element of I.
If every such f extends, then I is injective.
The proof extends a partial map maximally by Zorn's lemma and uses the criterion to show the maximal extension is total.

Over ℤ the ideals are (n), and the criterion says a map (n) → I extends to ℤ exactly when every element of I is divisible by n. Hence:

I injective over ℤ ⇔ I divisible

The standard examples

ℚ, ℚ/ℤ and the Prüfer groups ℤ(p^∞) are the injective abelian groups, and every injective abelian group is a direct sum of copies of them. ℤ itself is not injective, which is why Ext¹(−, ℤ) is interesting.

Section 03Enough injectives

That every module is a quotient of a projective is immediate. The dual statement — that every module embeds in an injective — requires work.

Stage 01Abelian groups firstEmbed M in a divisible group by presenting it as a quotient of a free group and enlarging ℤ to ℚ.
Stage 02Change ringsFor a Λ-module M, use the adjunction Hom_ℤ(Λ, −) to carry a divisible abelian group to an injective Λ-module.
Stage 03EmbedCompose to embed M in an injective Λ-module.
Stage 04IterateRepeating on the cokernel produces an injective resolution of arbitrary length.

Adjointness does the work

The functor Hom_ℤ(Λ, −) is right adjoint to restriction of scalars, and right adjoints of exact functors preserve injectives. This one observation converts the abelian-group case into the general case, and it is the first place adjointness earns its keep in the subject.

Section 04Injective hulls

Every module M has an injective hull: an injective E(M) containing M as an essential submodule, minimal among injectives containing M, and unique up to isomorphism over M.

No projective dual

Projective covers do not always exist — rings over which they do are called perfect. This asymmetry is one of the few places where the projective and injective theories genuinely diverge rather than merely mirror each other.

ReferenceFrequently asked questions

Why is Baer's criterion enough?

Because a maximal partial extension, produced by Zorn's lemma, can be shown total: if some element were outside its domain, the criterion applied to the ideal of ring elements carrying that element into the domain would extend it further, contradicting maximality.

Is ℚ/ℤ injective?

Yes — it is divisible, hence injective as an abelian group. It also serves as a dualizing object: Hom(−, ℚ/ℤ) is exact and faithful, which makes it the standard tool for character-module arguments and for proving flatness criteria.

Do I need injectives if I already have projectives?

For contravariant left exact functors and for right derived functors of covariant left exact functors, yes. Ext can be computed from either a projective resolution in the first variable or an injective resolution in the second, and the agreement of the two is a theorem worth knowing.

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ProvenanceSources and further reading

This page is an original KEVOS explanatory article. It presents the underlying mathematics — definitions, algorithms, complexity results and selection criteria — in KEVOS editorial voice. No text is reproduced from any copyrighted source. Where numerical tables are relevant, KEVOS links to live authoritative databases rather than republishing static values.

Handbook application: from concept to controlled practice

Purpose. This expanded section turns the original page into a practical handbook. It preserves the supplied material and adds a repeatable way to apply, check and review Injective Modules and Dualization. It does not replace a contract, legislation, a controlled standard, competent engineering judgement or specialist advice.

The operating aim is to turn a compact mathematical statement into a usable chain of definitions, claims, examples and checks. Read the original explanation first, then use the workflow and checks below to convert knowledge into evidence.

Treat Injective Modules and Dualization as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—injective, section, modules, criterion, enough—should be made explicit before any proof or computation begins. Record the ambient set or structure, the permitted operations and the equality or equivalence relation in use. A compact theorem often changes meaning when the base field, finiteness condition, commutativity assumption or direction of an action changes.

For a proof, write the hypotheses as a checklist and mark the line at which each one is used. For a computation, state the representation of the input, the arithmetic model, the termination condition and the output invariant. For a classification problem, distinguish existence from uniqueness and distinguish an object from its representation. These separations prevent a correct local calculation from being mistaken for the general result.

A useful worked example should be small enough to inspect completely but rich enough to exercise the main mechanism. Compute the result in two ways where practical: symbolically and by substitution, structurally and numerically, or directly and through a normal form. Then include one near-miss example in which a hypothesis fails. The contrast explains why the theorem is shaped as it is and gives the reader a diagnostic pattern for later problems.

Verification is part of the mathematics. Check domains and codomains, substitute proposed solutions, test identity and zero cases, compare dimensions or cardinalities, and confirm that maps respect the required operations. In numerical work, report precision, conditioning and a residual rather than digits alone. In algorithmic work, separate mathematical correctness from implementation complexity and resource limits.

Step-by-step operating method

Fix the setting. State the objects, ambient structure, notation and assumptions before manipulating symbols.
Separate claims. Distinguish definitions, hypotheses, conclusions, equivalent conditions and consequences.
Choose a method. Select proof, construction, calculation or algorithm according to the question actually asked.
Work a small case. Use the smallest non-trivial example to expose the mechanism and test edge behaviour.
Verify independently. Substitute back, check invariants, test boundary cases or use an alternative derivation.

Worked-example protocol

Illustrative method—not a source theorem. Start with a small admissible input and list the definitions it must satisfy. Carry out each transformation on a separate line, citing the property that permits it. Preserve exact values until approximation is necessary. At the end, verify the output against the original definition and one invariant such as dimension, degree, determinant, order, norm or residual. Then alter one hypothesis and observe which step ceases to be valid. This protocol creates a reusable example without inventing a theorem-specific numerical answer.

Stage	Record	Quality check
Input	Objects, domain, notation, assumptions	Every symbol is defined
Method	Permitted operation or cited result at each step	All hypotheses hold
Output	Exact result and representation	Correct type, domain and form
Verification	Substitution, invariant or alternative derivation	Independent agreement
Boundary test	Zero, identity, degenerate or failed hypothesis	Scope is understood

Common failure modes and recovery actions

1. Watch for

Using a theorem without checking every hypothesis.

Recovery: Return to the governing definition or requirement and restate the decision in one sentence.

2. Watch for

Treating a suggestive example as a proof of the general case.

Recovery: Separate evidence from assumption, assign an owner and set a date for validation.

3. Watch for

Changing notation or conventions part-way through an argument.

Recovery: Run a small counterexample, boundary test, pilot or independent check before proceeding.

4. Watch for

Hiding a division-by-zero, convergence, finiteness or commutativity assumption.

Recovery: Record the consequence, decision and rationale, then update the controlled baseline.

5. Watch for

Reporting a computed result without a residual, substitution or structural check.

Recovery: Escalate when the issue affects safety, compliance, acceptance, material value or an agreed tolerance.

Review checklist

Can every symbol be traced to a definition or prior result?
Which hypothesis does each major step use?
Does the method cover zero, identity, degenerate and boundary cases?
Can the conclusion be checked by a second representation or calculation?
Are mandatory requirements distinguished from recommendations and illustrative values?
Are sources, assumptions, units, dates and versions recorded closely enough to reproduce the decision?
Have safety, legal, ethical, stakeholder and operational consequences been considered at the appropriate level?
Is there a named owner and a trigger for review, escalation, change or retirement?

Questions for deeper application

What is the most important distinction a practitioner must preserve when applying Injective Modules and Dualization?