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A module is projective iff it is a direct summand of a free module.
Projective is a summand of free: An -module is projective if and only if there exists a free module and an -module such that
as a direct sum.
This characterization is often the most practical: projective modules are precisely the modules that can be “split off” from free modules (equivalently, retracts of free modules).
An -module is projective if for every surjective homomorphism and every homomorphism (see module homomorphisms), there exists a homomorphism such that .
A left -module is free if it has a basis , i.e. every element of admits a unique expression as a finite -linear combination of elements of .
Free modules are characterized by the universal property of free modules: functions from a basis extend uniquely to module homomorphisms. Over a field, free modules coincide with vector spaces and their bases agree with linear-algebra bases.
Given a family of -modules , their direct sum is the set
with coordinatewise addition and scalar multiplication. It is naturally a submodule of the direct product, which itself is modeled on the Cartesian product of sets.
The direct sum is characterized by the universal property of the direct sum: maps out of a direct sum are uniquely determined by maps out of each summand, subject to finite support.