Linear Algebra over a Ring/Free modules and matrices

Definition (free module):

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

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

Proposition (basis of a free module):

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

By abuse of notation, we will write instead of . Hence, the above proposition implies that we may denote an element as a sum


where only finitely many are nonzero.

Proof: Let be any function that is everywhere zero except on finitely many entries, and let . Then we have