Definition
A left module Q over the ring R is injective if it satisfies one (and therefore all) of the following equivalent conditions:
- If Q is a submodule of some other left R-module M, then there exists another submodule K of M such that M is the internal direct sum of Q and K, i.e. Q + K = M and Q ∩ K = {0}.
- Any short exact sequence 0 →Q → M → K → 0 of left R-modules splits.
- If X and Y are left R-modules and f : X → Y is an injective module homomorphism and g : X → Q is an arbitrary module homomorphism, then there exists a module homomorphism h : Y → Q such that hf = g, i.e. such that the following diagram commutes:
- The contravariant functor Hom(-,Q) from the category of left R-modules to the category of abelian groups is exact.
Injective right R-modules are defined in complete analogy.
Read more about this topic: Injective Module
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