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 $R\langle S\rangle$, 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

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

By abuse of notation, we will write $s$ instead of $f_{s}$. Hence, the above proposition implies that we may denote an element $m\in R\langle S\rangle$ 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 $s_{1},\ldots ,s_{n}=\{s\in S|f(s)\neq 0\}$. Then we have

$f=\sum _{k=1}^{n}f(s_{k})s_{k}$. $\Box$