Injective Module

An injective module is a mathematical structure in the field of algebra that behaves like a “flexible container” for other modules, allowing certain types of extensions to happen seamlessly. Specifically, it can absorb embeddings (injections) from other modules without losing essential information, meaning any partial structure can be extended to the whole module without inconsistency. This property makes injective modules important for understanding how modules can be embedded into larger structures and analyzed using extension techniques. In simpler terms, injective modules are those that can “complete” or “fill in” parts of algebraic structures in a consistent and reliable manner.