Linear Algebra over a Ring/Free modules and matrices

Definition (free module):

Let $R$ be a ring, and let $S$ be an arbitrary set. Then the free $R$-module over $S$, denoted $R⟨S⟩$, is defined to be the $R$-module whose elements are functions

$f:S\to R$

which are zero everywhere on $S$ except on finitely many elements, together with pointwise addition and scalar multiplication.

Proposition (basis of a free module):

Let $R$ be a ring, and let $S$ be a set. Then a basis for the free $R$-module over $s$ is given by the functions

$f_s:S\to R,f(t):=\{\begin{array}{cc}1& t=s\\ 0& \text{otherwise}.\end{array}$

By abuse of notation, we will write $s$ instead of $f_s$. Hence, the above proposition implies that we may denote an element $m\in R⟨S⟩$ as a sum

$m=\sum _{s\in S}{a}_{s}s$,

where only finitely many $a_s$ are nonzero.

Proof: Let $f:S\to R$ be any function that is everywhere zero except on finitely many entries, and let $s_1,\dots ,s_n=\{s\in S|f(s)\ne 0\}$. Then we have

$f=\sum _{k=1}^{n}f(s_k)s_k$. $◻$