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 \(\mathbb R\)/\(\mathbb Z\).

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