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 ∈ S}
*r*_{s} *s*, or *X* is a direct sum
*X* = ⊕_{s ∈ 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* \(\circ\) *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 *: *G* →
*G*', 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* : *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 \(\mathbb Z\)-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* : *B*
→ *C* and homomorphism *γ* : *X* →
*C* of *R*-modules, there is a homomorphism *β* :
*X* → *B*, 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}
*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 *μ *: *A* → *B* and
homomorphism *α* : *A* → *X* of *R*-modules,
there is a homomorphism *β* : *B* → *X*,
with *βμ* = *α* ("*α* can
be extended to *β*").

$ __Divisible__: *X* is divisible
if for all *x* ∈ *X* and non-zero *λ* ∈
\(\mathbb R\), ∃ *y* ∈ *X* (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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send feedback and suggestions to bombelli at olemiss.edu – modified 15 apr 2019