Projective and Injective Modules - Key Properties

Understanding the characterizations and properties of projective and injective modules is essential for computational homological algebra.

Theorem3.5Characterization of Projective Modules

For an $R$ -module $P$ , the following are equivalent:

$P$ is projective
$\text{Ext}^1_R(P, M) = 0$ for all $R$ -modules $M$
Every surjection onto $P$ splits
$P$ is a direct summand of a free module

Proof

(1) ⇒ (3): Let $f: M \twoheadrightarrow P$ be surjective. Consider the exact sequence $0 \to \ker(f) \to M \xrightarrow{f} P \to 0$ . By projectivity, the identity $\text{id}_P: P \to P$ lifts to $s: P \to M$ with $f \circ s = \text{id}_P$ , so the sequence splits.

(3) ⇒ (4): Take a free module $F$ surjecting onto $P$ . By (3), $P$ is a direct summand of $F$ .

(4) ⇒ (1): Free modules are projective, and direct summands of projective modules are projective.

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Theorem3.6Baer's Criterion for Injectivity

An $R$ -module $I$ is injective if and only if for every ideal $\mathfrak{a} \subseteq R$ and every homomorphism $\phi: \mathfrak{a} \to I$ , there exists an extension $\tilde{\phi}: R \to I$ .

Remark

Baer's Criterion is powerful because it reduces the lifting condition from all modules to just ideals. This makes verifying injectivity much more tractable in practice.

Definition3.7Essential Extension

A submodule $N \subseteq M$ is essential if for every non-zero submodule $K \subseteq M$ , we have $N \cap K \neq 0$ . An injective module $E(M)$ containing $M$ as an essential submodule is called an injective envelope or injective hull of $M$ .

Theorem3.8Existence and Uniqueness of Injective Hull

Every $R$ -module $M$ has an injective hull $E(M)$ , which is unique up to isomorphism over $M$ .

ExampleInjective Hull of $\mathbb{Z}/n\mathbb{Z}$

The injective hull of $\mathbb{Z}/n\mathbb{Z}$ is the Prüfer group: $\mathbb{Z}(p^\infty) = \{x \in \mathbb{Q}/\mathbb{Z} : \text{order of } x \text{ is a power of } p\}$

where $n = p^k$ for prime $p$ . More generally, $E(\mathbb{Z}/n\mathbb{Z})$ is a direct sum of Prüfer groups for each prime dividing $n$ .

Definition3.9Pure Submodule

A submodule $N \subseteq M$ is pure if for every $R$ -module $K$ , the natural map $N \otimes_R K \to M \otimes_R K$ is injective. Equivalently, every finite system of equations over $N$ that has a solution in $M$ has a solution in $N$ .

Theorem3.10Pure-Injective Modules

A module $M$ is pure-injective if every pure embedding $M \hookrightarrow N$ splits. Pure-injective modules form an important class generalizing injective modules, with applications in model theory.