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Coinduction sends injective abelian groups to injective modules

Statement

Let $R$ be a unital ring and $D$ an injective abelian group. Give ${Hom}_{Z} (R, D)$ the left $R$ -action $(r φ) (s) = φ (s r) .$ Then ${Hom}_{Z} (R, D)$ is an injective left $R$ -module.

Facts & Assumptions

Given: A unital ring $R$ and an injective abelian group $D$ .

[F1]

Injectivity is extension of homomorphisms along monomorphisms (Injective modules and the extension property).

[F2]

$Hom$ groups use pointwise addition and maps induced by composition (The abelian group ${Hom}_{R} (M, N)$ and maps induced by pre- and postcomposition).

Proof

technique · direct

1.1

The formula $(r φ) (s) = φ (s r)$ satisfies $r_{1} (r_{2} φ) = (r_{1} r_{2}) φ$ , the distributive laws, and $1 φ = φ$ , so it defines a left $R$ -module structure.

givenF2algebra

2.1

For every left $R$ -module $M$ , evaluation at $1$ defines $Θ_{M} : {Hom}_{R} (M, {Hom}_{Z} (R, D)) \to {Hom}_{Z} (M, D), Θ_{M} (F) (m) = F (m) (1) .$

step 1.1F2

3.1

The inverse sends $g : M \to D$ to $\hat{g} (m) (r) = g (r m)$ . Indeed, $\hat{g}$ is $R$ -linear because $\hat{g} (s m) (r) = g (r s m) = (s \hat{g} (m)) (r)$ , and evaluation at $1$ and the unit law show that the two constructions are inverse.

step 1.1step 2.1algebra

3.2

Let $u : A \to B$ be a monomorphism and $F : A \to {Hom}_{Z} (R, D)$ . Under $Θ_{A}$ , it corresponds to a group homomorphism $g : A \to D$ , which extends along the underlying subgroup inclusion to $\tilde{g} : B \to D$ because $D$ is injective.

givenstep 2.1F1

4.1

The inverse construction of step 3.1 turns $\tilde{g}$ into an $R$ -linear extension $\hat{\tilde{g}} : B \to {Hom}_{Z} (R, D)$ of $F$ . Hence the coinduced module is injective by [F1].

step 3.1step 3.2F1∎

Depends on

Used by

Module categories have enough injectives Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 14 results over 11 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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)