In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, by keeping some of the main properties of free modules. Various equivalent characterizations of these modules appear below.
Every free module is a projective module, but the converse fails to hold over some rings, such as Dedekind rings that are not principal ideal domains. However, every projective module is a free module if the ring is a principal ideal domain such as the integers, or a polynomial ring (this is the Quillen–Suslin theorem).
Projective modules were first introduced in 1956 in the influential book Homological Algebra by Henri Cartan and Samuel Eilenberg.
Definitions
Lifting property
The usual category theoretical definition is in terms of the property of lifting that carries over from free to projective modules: a module P is projective if and only if for every surjective module homomorphism and every module homomorphism , there exists a module homomorphism such that . (We don't require the lifting homomorphism h to be unique; this is not a universal property.)
The advantage of this definition of "projective" is that it can be carried out in categories more general than module categories: we don't need a notion of "free object". It can also be dualized, leading to injective modules. The lifting property may also be rephrased as every morphism from to factors through every epimorphism to . Thus, by definition, projective modules are precisely the projective objects in the category of R-modules.
Split-exact sequences
A module P is projective if and only if every short exact sequence of modules of the form
is a split exact sequence. That is, for every surjective module homomorphism there exists a section map, that is, a module homomorphism such that f h = idP . In that case, is a direct summand of B, h is an isomorphism from P to , and is a projection on the summand . Equivalently,
Direct summands of free modules
A module P is projective if and only if there is another module Q such that the direct sum of P and Q is a free module.
Exactness
An R-module P is projective if and only if the covariant functor is an exact functor, where is the category of left R-modules and Ab is the category of abelian groups. When the ring R is commutative, Ab is advantageously replaced by in the preceding characterization. This functor is always left exact, but, when P is projective, it is also right exact. This means that P is projective if and only if this functor preserves epimorphisms (surjective homomorphisms), or if it preserves finite colimits.
Dual basis
A module P is projective if and only if there exists a set and a set such that for every x in P, fi  (x) is only nonzero for finitely many i, and .
Elementary examples and properties
The following properties of projective modules are quickly deduced from any of the above (equivalent) definitions of projective modules: Direct sums and direct summands of projective modules are projective. If is an idempotent in the ring R, then Re is a projective left module over R.
Relation to other module-theoretic properties
The relation of projective modules to free and flat modules is subsumed in the following diagram of module properties:
The left-to-right implications are true over any ring, although some authors define torsion-free modules only over a domain. The right-to-left implications are true over the rings labeling them. There may be other rings over which they are true. For example, the implication labeled "local ring or PID" is also true for polynomial rings over a field: this is the Quillen–Suslin theorem.
Projective vs. free modules
Any free module is projective. The converse is true in the following cases: if R is a field or skew field: any module is free in this case. if the ring R is a principal ideal domain. For example, this applies to (the integers), so an abelian group is projective if and only if it is a free abelian group. The reason is that any submodule of a free module over a principal ideal domain is free. if the ring R is a local ring. This fact is the basis of the intuition of "locally free = projective". This fact is easy to prove for finitely generated projective modules. In general, it is due to ; see Kaplansky's theorem on projective modules.