Why are free modules called "free"?

Question

Let $R$ be a ring (not necessarily commutative) with multiplicative identity.

A $R$-module $M$ is called free if $M$ has a linearly independent generating set $\beta\subseteq M$.

That is, for any $m\in M$, $$m=\sum_{i=1}^{n} r_i b_i$$ for some $n\in \Bbb{N}^+$ and $r_1, r_2,...,r_n\in R$ and $b_1, b_2,...,b_n\in \beta$ and if $\sum_{i=1}^{n} r_i b_i=0$, where $b_1, b_2,..., b_n$ are distinct, then $r_1=r_2=\cdots =r_n=0$.

I can't feel the sense of the "free".

My guess is that the coefficients in the linear combination of the basis elements is "arbitrary". But it seems not to be a satisfactory explanation. Because in the non-free $\Bbb{Z}$-module $\Bbb{Z}_2\oplus \Bbb{Z}_3$, the coefficients of the generating set $\{(1,0),(0,1)\}$ is also arbitrary.

Answer 1

Free is in the sense of "having no non-trivial relations", or alternatively, in the sense of coming "for free" (i.e. without any additional work necessary) from a set.

For example, in the $\mathbb{Z}$-module $\mathbb{Z}/3\mathbb{Z}$, even though $\{\overline{1}\}$ is a generating set, it has the property that $$3\cdot\overline{1}=0\cdot\overline{1}$$ whereas in a free $\mathbb{Z}$-module $M$, if $\{m_\alpha\}$ is a generating set then the only way $\sum_{i=1}^n a_im_i=0_M$ to be true for some $a_i\in\mathbb{Z}$ and $m_i\in \{m_\alpha\}$ is if every $a_i=0$.

In the other sense, given a set $S$, we can form the free $\mathbb{Z}$-module on $S$ by simply saying $$F(S)=\{\textstyle\sum_{i=1}^na_is_i:a_i\in\mathbb{Z},s_i\in S\}$$ whereas to make a non-free module requires the extra act of imposing relations between such elements.

Free modules are just one example of a free object (Wikipedia link). Take a look at the math.SE threads What are Free Objects? and definition of a free object in a category.