Free module

A free object (a free algebra) in the variety of modules over a fixed ring $R$. If $R$ is associative and has a unit, then a free module is a module with a basis, that is, a linearly independent system of generators. The cardinality of a basis of a free module is called its rank. The rank is not always defined uniquely, that is, there are rings over which a free module can have two bases consisting of a different number of elements. This is equivalent to the existence over $R$ of two rectangular matrices $A$ and $B$ for which

$$AB=I_m,\quad BA=I_n,\quad m\neq n,$$

where $I_m$ and $I_n$ denote the unit matrices of orders $m$ and $n$, respectively. However, non-uniqueness holds only for finite bases; if the rank of a free module is infinite, then all bases have the same cardinality. In addition, over rings that admit a homomorphism into a skew-field (in particular, over commutative rings), the rank of a free module is always uniquely defined.

A ring $R$, considered as a left module over itself, is a free module of rank 1. Every left free module is a direct sum of free modules of rank 1. Every module $M$ is representable as a quotient module $F_0/H_0$ of a free module $F_0$. The submodule $H_0$ is, in turn, representable as a quotient module $F_1/H_1$ of a free module $F_1$. By continuing this process one obtains the exact sequence

$$\dotsb\to F_2\to F_1\to F_0\to M\to0,$$

which is called the free resolution of $M$. Skew-fields can be characterized as rings over which all modules are free. Over a principal ideal domain a submodule of a free module is free. Near to free modules are projective modules and flat modules (cf. Projective module; Flat module).

References

[1]	P.M. Cohn, "Free rings and their relations" , Acad. Press (1971)
[2]	S. MacLane, "Homology" , Springer (1963)