Injective module

An injective module has the extension property for homomorphisms defined on submodules.

The divisible group $D=\mathbb{Q}/\mathbb{Z}$ is injective: every homomorphism $n\mathbb{Z}\to D$ extends to $\mathbb{Z}$ by choosing an $n$th division of its value at $n$, and the Baer criterion applies. It is a cogenerator because any nonzero element generates a cyclic subgroup admitting a character nonzero on that element, and injectivity extends that character to the whole group. Consequently a map $u:T\to U$ of abelian groups is injective exactly when restriction $U^{\ast }\to T^{\ast }$ is surjective: injectivity gives extension of characters, while a nonzero kernel element is detected by a character that cannot extend along $u$.

For a Noetherian ring, injectivity of a module can be tested at all prime localizations. By the Baer criterion, test ${\mathrm{Ext}}_R^1(R/J,M)$ for every ideal $J$. Localization of Ext over a Noetherian ring identifies its localization with the corresponding ideal test over $R_P$. Every ideal of $R_P$ is extended from its contraction, and localization detects zero elements applies to the resulting Ext module even when it is not finitely generated. These observations prove both directions.

This criterion characterizes an injective module using extension from ideals. A left $R$-module is injective exactly when every homomorphism from a left ideal of $R$ extends to a homomorphism from $R$.

A ring is left Noetherian exactly when every direct sum of injective left modules is injective.

Over an integral domain, a module $M$ is divisible when $rM=M$ for every nonzero $r$. A divisible group is precisely a divisible $\mathbb{Z}$-module, and over a principal ideal domain divisibility is equivalent to injectivity.

The injective hull of a module $M$ is a minimal injective extension of $M$ in which $M$ is an essential submodule.

For a domain satisfying the left Ore condition, its division ring of fractions $Q$ is an injective essential extension of the left regular module. For a homomorphism $f:I\to Q$ on a nonzero left ideal, choose $a\in I\setminus \{0\}$. A common left multiple $sx=ta$ gives $sf(x)=tf(a)$, hence $f(x)=x{a}^{-1}f(a)$; right multiplication by $a^{-1}f(a)$ extends it. The Baer criterion proves injectivity. Clearing a left denominator proves essentiality. The same calculation gives ${\mathrm{End}}_R({}_{R}Q)\cong Q^{\mathrm{op}}$, with the opposite ring required because right multiplications compose in reversed order.

The injective hull of a uniform module is uniform. An injective endomorphism of a uniform injective module splits its image, so its complementary summand must vanish and it is an automorphism. Consequently all nonunits are precisely the noninjective endomorphisms. Their nonzero kernels are essential; intersection of two such kernels is nonzero, making their sum noninjective. Composition on either side preserves noninjectivity, unless the other map is a unit, which merely transports the kernel. Thus the nonunits form the unique maximal ideal, and the quotient is a division ring.