Injective Modules, Injective Embeddings and Divisible Groups
Injective Modules, Injective Embeddings and Divisible Groups: core definitions, structural results and verification methods in abstract algebra.
How the topic fits together
Injective Modules
This source section supplies the definitions, constructions and formal results used in this article. The sequence of results is preserved so later claims are not detached from their prerequisites.
Embedding into an Injective Module
This source section supplies the definitions, constructions and formal results used in this article. The sequence of results is preserved so later claims are not detached from their prerequisites.
Divisible abelian groups appendix
This source section supplies the definitions, constructions and formal results used in this article. The sequence of results is preserved so later claims are not detached from their prerequisites.
Core definitions and structural results
The following cards retain the mathematical content of the source while condensing long proofs into verification strategies. Numerical examples are treated as examples, not universal requirements.
Definition
The R-module E is injective if given g : N →E, and f : N →M injective, there exists h : M →E (not necessarily unique) such that g = hf. We sometimes say that one has “lifted” g to h. As with projectives, there are several equivalent ways to characterize an injective module.
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
Theorem
The following conditions on the R-module E are equivalent. (1) E is injective. (2) The functor HomR( , E) is exact. (3) Every exact sequence 0 →E →M →N →0 splits.
Proof / verification strategy: Work directly from closure, inverses, subgroup tests and the defining action or presentation. Use a small cyclic or permutation model to verify the direction of the argument.
Proposition
A direct product j Ej of modules is injective iffeach Ej is injective. Consequently, a finite direct sum is injective iffeach summand is injective.
Proof / verification strategy: Use subgroup generation, coset/orbit counting and divisibility of orders. For cyclic groups, reduce the claim to arithmetic on exponents modulo the group order.
Baer’s Criterion The R-module E is injective if and only if every R-homomorphism
Baer’s Criterion The R-module E is injective if and only if every R-homomorphism f : I →E, where I is a left ideal of R, can be extended to an R-homomorphism h : R →E.
Proof / verification strategy: Work directly from closure, inverses, subgroup tests and the defining action or presentation. Use a small cyclic or permutation model to verify the direction of the argument.
Definition
Let R be an integral domain. The R-module M is divisible if each y ∈M can be divided by any nonzero element r ∈R, that is, there exists x ∈M such that rx = y. For example, the additive group of rational numbers is a divisible abelian group, as is Q/Z, the rationals mod 1. The quotient field of any integral domain (regarded as an abelian group) is divisible.
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
Proposition
If R is any integral domain, then an injective R-module is divisible. If R is a PID, then an R-module is injective if and only if it is divisible.
Proof / verification strategy: Work directly from closure, inverses, subgroup tests and the defining action or presentation. Use a small cyclic or permutation model to verify the direction of the argument.
Proposition
Every abelian group can be embedded in a divisible abelian group.
Proof / verification strategy: Work directly from closure, inverses, subgroup tests and the defining action or presentation. Use a small cyclic or permutation model to verify the direction of the argument.
Comments In (10.7.1), we used Q as a standard divisible abelian group. It would be
Comments In (10.7.1), we used Q as a standard divisible abelian group. It would be very desirable to have a canonical injective R-module. First, we consider H = HomZ(R, A), the set of all abelian group homomorphisms from the additive group of the ring R to the abelian group A. If we are careful, one can make this set into a left R-module.
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
Proposition
If E is a divisible abelian group, then HomZ(R, E) is an injective left R-module.
Proof / verification strategy: Work directly from closure, inverses, subgroup tests and the defining action or presentation. Use a small cyclic or permutation model to verify the direction of the argument.
Proposition
Let H be a group defined by generators b1, b2, . . . and relations pb1 = 0, pb2 = b1, . . . , pbr+1 = br, . . .. Then H is isomorphic to Z(p∞).
Proof / verification strategy: Construct the natural group homomorphism, identify its kernel and image, and then use quotient structure to prove the claimed isomorphism or correspondence.
Proposition
Let G be a divisible abelian group. Then its torsion subgroup T is also divisible. Moreover, G can be written as T ⊕D, where D is torsion-free and divisible.
Proof / verification strategy: Work directly from closure, inverses, subgroup tests and the defining action or presentation. Use a small cyclic or permutation model to verify the direction of the argument.
Proposition
If G is a divisible, torsion-free abelian group, then G is isomorphic to a direct sum of copies of Q.
Proof / verification strategy: Construct the natural group homomorphism, identify its kernel and image, and then use quotient structure to prove the claimed isomorphism or correspondence.
Proposition
Let G and H be divisible abelian p-groups. Then any isomorphism ϕ of G[p] and H[p] can be extended to an isomorphism ψ of G and H.
Proof / verification strategy: Construct the natural group homomorphism, identify its kernel and image, and then use quotient structure to prove the claimed isomorphism or correspondence.
Theorem
An abelian group G is divisible if and only if G is a direct sum of copies of Q and quasicyclic groups.
Proof / verification strategy: Work directly from closure, inverses, subgroup tests and the defining action or presentation. Use a small cyclic or permutation model to verify the direction of the argument.
Quick-reference relationships
Problem-solving workflow
Identify the ambient group
State the operation, identity, inverses and whether commutativity is available.
Locate the relevant subgroup structure
Check generated subgroups, normality, cosets, stabilisers or direct factors before applying a theorem.
Use the correct counting or mapping tool
Choose coset counting, orbit counting, a homomorphism, a quotient or a group action as appropriate.
Check hypotheses explicitly
Finite-order assumptions, normality, prime-power divisibility and coprimality conditions are not interchangeable.
Translate the result back to structure
Interpret the result as a statement about subgroup size, quotient structure, action, decomposition or presentation.
Verify with a small model
Use a cyclic, symmetric, dihedral or modular example to test the logic without treating the example as the theorem.
Worked-solution emphasis from the supplied source
The supplied worked solutions for this section repeatedly test quotient, basis, module, homomorphism, exact, projective, injective, Tor. These checks are used here as verification themes rather than copied as answer text.
Common mistakes and boundary conditions
- Treating left and right cosets as identical without normality.
- Assuming the converse of a subgroup-order divisibility result.
- Confusing the order of a group with the order of one of its elements.
- Using quotient multiplication before checking that the subgroup is normal.
- Forgetting the coefficient ring when comparing modules.
- Assuming tensor product preserves every exact sequence.
Verification checklist
- State the ambient algebraic structure and operation before applying a theorem.
- Record every hypothesis that controls the result: finiteness, commutativity, normality, primality, separability, exactness or other section-specific conditions.
- Distinguish a definition from a theorem that follows from it.
- Check whether a map is well-defined before using its kernel, image, inverse or induced map.
- Use a concrete example only as a check; do not promote an illustrative value or pattern to a universal rule.
- When a quotient, localisation or extension is constructed, identify the canonical map and what becomes equal, invertible or fixed.
Source coverage map
| Source section | Subject | PDF pages analysed |
|---|---|---|
| 10.6 | Injective Modules | 211–213 |
| 10.7 | Embedding into an Injective Module | 214–216 |
| A10 | Divisible abelian groups appendix | 224–225 |
Related Mathematics pages
Source note: synthesised from the supplied abstract-algebra PDF. The complete 298-page file, including diagrams and worked solutions, was reviewed. Source-identifying author and bibliographic personal details are intentionally omitted. Formal proofs are condensed; the page does not claim requirements or values not supported by the supplied mathematics.