Types of Modules

Free Modules
$Def 1: An R-module X is free if it has a basis S, i.e., one can write uniquely any xX as x = ∑sS rs s, or X is a direct sum X = ⊕sS R.$ Def 2: Given a set S, an R-module X and a function a : SX, we say that (X, a) is a free module on S if, for any R-module Y and function g : SY, there is a unique homomorphism m : XY, such that g = m $$\circ$$ a.
* Remark: In terms of the general definition above for groups, the homomorphism m can be constructed by linearity by knowing the map g on the base S.
* Properties:
- Meaning: If (X, a) is a free R-module on S, then a is 1-1 and a(S) generates G;
- Uniqueness: If (X, a) and (X', a') are free R-modules on S, there exists a unique isomorphism f : GG', such that f $$\circ$$ a = a';
- Existence: For any set S, there exists a free R-module (X, a) on S; It can be generated as the group of functions f : SR such that f(s) ≠ 0 only for finitely many values of s;
- Relationships: For any R-module Y there is a free R-module (X, a), and an epimorphism m : XY; In other words, every module is a quotient of a free one.
* Free Abelian group: It is a free $$\mathbb Z$$-module, and all the above can be said for a free abelian group substituting $$\mathbb Z$$ for R; A free abelian group is always torsion-free.
* Finitely generated: The number of elements in a base is independent of the choice of base; Any subgroup then is also a finitely generated free abelian group.

Projective Modules
$Def: X is projective if for any epimorphism ε : BC and homomorphism γ : XC of R-modules, there is a homomorphism β : XB, with εβ = γ. * Relationships: Every free module is projective; any projective module is a direct summand in a free module. * Example: $$\mathbb Z$$2 and $$\mathbb Z$$3 are projective $$\mathbb Z$$6 modules ($$\mathbb Z$$2 ⊕ $$\mathbb Z$$3 = $$\mathbb Z$$6). * Properties: The direct sum ⊕i Pi is projective iff Pi is projective for all i. Other Types > s.a. modules [examples]. * Over a ring with identity: If R has an identity e, we require that ex = x, for all xX.$ Injective: X is injective if for any monomorphism μ : AB and homomorphism α : AX of R-modules, there is a homomorphism β : BX, with βμ = α ("α can be extended to β").
\$ Divisible: X is divisible if for all xX and non-zero λ ∈ $$\mathbb R$$, ∃ yX (not necessarily unique) such that λy = x; Examples: $$\mathbb Q$$ as a $$\mathbb Z$$-module (y is unique); $$\mathbb Q$$/$$\mathbb Z$$ as a $$\mathbb Z$$-module (y not unique); similarly $$\mathbb R$$ and $$\mathbb R$$/$$\mathbb Z$$.