Injective module

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

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 $r$. 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.