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__:
⊕_{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 *β**μ* = *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 *λ**y* = *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 14
feb
2016