Injective Module - Definition

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 QK = {0}.
  • Any short exact sequence 0 →QMK → 0 of left R-modules splits.
  • If X and Y are left R-modules and f : XY is an injective module homomorphism and g : XQ is an arbitrary module homomorphism, then there exists a module homomorphism h : YQ 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.

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