An injective module is a type of module that satisfies the property that any homomorphism from a submodule can be extended to the entire module. This means that if you have a module $M$ and an injective module $I$, any morphism defined on a submodule of $M$ can be lifted to $I$. Injective modules play a crucial role in homological algebra, particularly in the context of Ext and Tor functors, where they help characterize projective resolutions and modules that are 'nice' with respect to exact sequences.
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