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 x X as x = _{s in S} r_s s, or X is a direct sum X = _{s in S} R.
$ Def 2: Given a set S, an R-module X and a function a: S → X, we say that (X, a) is a free module on S if, for any R-module Y and function g: S → Y, there is a unique homomorphism m: X → Y, such that g = m 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 : G → G', such that f 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: S → R 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: X → Y; In other words, every module is a quotient of a free one.
* Free Abelian group: It is a free Z-module, and all the above can be said for a free abelian group substituting 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 : B → C and homomorphism : X → C of R-modules, there is a homomorphism : X → B, with = .
* Relationships: Every free module is projective; any projective module is a direct summand in a free module.
* Example: Z₂ and Z₃ are are projective Z₆ modules (Z₂ Z₃ = Z₆).
* Properties: _i P_i is projective iff P_i 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 x X.
$ Injective: X is injective if for any monomorphism : A → B and homomorphism : A → X of R-modules, there is a homomorphism : B → X, with = (" can be extended to ").
$ Divisible: X is divisible if for all x X and nonzero R, y X (not necessarily unique) such that y = x; Examples: Q as a Z-module (y is unique); Q/Z as a Z-module (y not unique); similarly R and R/Z.

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Send feedback and suggestions to bombelli at olemiss.edu – Modified 5 jul 2008