Projective, injective, and flat modules

A left module $P$ is projective if every map from $P$ into the target of a surjection lifts to its source. Equivalently, $\operatorname{Hom}_R(P,-)$ is exact, rather than merely left exact. Free modules have this property because images of basis elements can be lifted individually. A direct summand of a free module remains projective. Conversely, choose a free module surjecting onto $P$ and lift its identity; the resulting section exhibits $P$ as a direct summand.

This characterization does not require a finite basis. It also explains why projective is weaker than free over a general ring: a summand can fail to admit a global basis. A short exact sequence ending in a projective module splits by lifting that module's identity. The quantifier matters. Splitting one convenient sequence is not the definition; projectivity concerns every surjection and every compatible map, or equivalently every short exact sequence having that module as quotient.

A left module $J$ is injective if a map from a submodule into $J$ extends to the containing module. The contravariant functor $\operatorname{Hom}_R(-,J)$ is then exact. Every short exact sequence beginning with $J$ splits, by extending its identity to a retraction. Reversing arrows is essential: a projective condition at the quotient cannot be substituted for an injective condition at the submodule simply because both statements mention a splitting.

Over integers, injective modules are exactly divisible abelian groups. The rationals and the rationals modulo integers are standard examples; division is solvable inside these groups, although uniqueness is unnecessary. This criterion is specific to the integer ring, not a characterization of injectives over arbitrary rings. Categories of left modules have enough injectives, so every module embeds in one. An arbitrary abelian category need not have enough injectives or enough projectives; resolution constructions must explicitly assume the relevant existence property.

Tensor product is always right exact. For a right module $F$, flatness says that $F\otimes_R-$ also preserves injections of left modules, hence all short exact sequences. Every right projective module is flat because it is a summand of a free module, and tensoring with a free module forms a direct sum. Flatness does not generally supply lifts of maps; it only controls how relations change after tensoring, which is weaker than the projective lifting condition.

For a commutative ring, a finitely presented flat module is projective. Omitting finite presentation changes the statement. The integer module $\mathbb Q$ is flat because it is a localization, but it is not projective: projective integer modules are free, and a nonzero free abelian group is not divisible. Thus localizing can remove torsion without creating a basis suitable for arbitrary lifting. When selecting a resolution, decide whether the calculation needs exact tensoring, exact Hom, or both before choosing its building blocks.