Injective Modules over a Principal Ideal Domain

Executive summary

Injective means divisible

Over a principal ideal domain, Baer's criterion collapses to a single condition: a module is injective exactly when it is divisible, meaning every element can be divided by every non-zero ring element. For abelian groups this classifies the injectives completely as direct sums of copies of ℚ and of Prüfer groups, and it gives every module an injective resolution of length at most one — the mirror of the projective situation.

Learning objectives

Characterise injectivity over a PID as divisibility.
Classify the injective abelian groups.
Construct a length-one injective resolution.
Explain the symmetry with the projective case and where it breaks.

Section 01Divisibility

A module D over a domain R is divisible when for every d ∈ D and non-zero r ∈ R there is d′ with rd′ = d. Over a PID this is exactly injectivity, by Baer's criterion applied to the ideals (r).

Divisible and non-divisible abelian groups
Group	Divisible?	Injective?
ℚ	Yes	Yes
ℚ/ℤ	Yes	Yes
ℤ(p^∞) (Prüfer)	Yes	Yes
ℝ, ℂ	Yes	Yes
ℤ	No	No
ℤ/nℤ	No	No — but injective as a ℤ/nℤ-module when n is squarefree

Injectivity depends on the ring, not the group

ℤ/pℤ is not injective over ℤ, but it is injective over ℤ/pℤ. Whenever the phrase ‘injective module’ appears, the ring must be named — the property is not intrinsic to the underlying abelian group.

Section 02Classification over ℤ

Every divisible abelian group is a direct sum of copies of ℚ and of Prüfer groups ℤ(p^∞), and the multiset of summands is an invariant. The torsion-free part is a ℚ-vector space; the p-primary part is a sum of Prüfer groups.

D ≅ ℚ^(I) ⊕ ⊕_p ℤ(p^∞)^(J_p)

The Prüfer group

ℤ(p^∞) is the union of the cyclic groups of order pⁿ, equivalently the p-primary part of ℚ/ℤ. It is the smallest injective containing ℤ/pℤ, so it is the injective hull of ℤ/pℤ — the standard first example of a hull.

Section 03Short injective resolutions

AlgorithmThe canonical injective resolution over a PIDin: a module over a PID → out: a length-one injective resolution

Embed M in an injective module I⁰ — over ℤ, embed in a divisible group.
Set I¹ = I⁰/M.
I¹ is divisible, being a quotient of a divisible module. This is the step that fails over a general ring.
So 0 → M → I⁰ → I¹ → 0 is an injective resolution of length 1.
Hence Extⁿ(−, M) = 0 for n ≥ 2, agreeing with the projective computation.

Quotients of divisibles are divisible

This closure property is what makes the resolution stop. It is the injective mirror of ‘submodules of free modules are free’, and both are special to hereditary rings.

ReferenceFrequently asked questions

Why is ℚ/ℤ so useful?

Because it is injective and cogenerates: a module is zero if and only if its dual into ℚ/ℤ is zero. That makes Hom(−, ℚ/ℤ) an exact faithful functor, which converts statements about a module into dual statements that are often easier to check — flatness criteria being the standard use.

Is a divisible group ever free?

Only the zero group. A non-zero free abelian group has elements not divisible by 2, so divisibility and freeness are incompatible except trivially. This is a compact illustration of how far the projective and injective worlds sit apart.

Does divisible imply injective over any domain?

Over a Dedekind domain, yes. Over a general domain, divisibility is necessary but not sufficient; injectivity is the stronger condition, and Baer's criterion is what must be checked.

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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 over a Principal Ideal Domain. 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 over a Principal Ideal Domain as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—injective, modules, over, section, divisibility—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 over a Principal Ideal Domain?