Module

   For r,s in R, x in M, (rs)x = r(sx)
   For r,s in R, x in M, (r+s)x = rx+sx
   For r in R, x,y in M, r(x+y) = rx+ry
   For x in M, 1x = x

A right R-module is defined similarly, only the ring acts on the right. The two are easily interchangeable.

The action of an element r in R is defined to be the map that sends each x to rx (or xr), and is necessarily an endomorphism of M. The set of all endomorphisms of M is denoted End(M) and forms a ring under addition and composition, so the above actually defines a homomorphism from R into End(M).

This is called a representation of R over M, and is called faithful if and only if the map is one-to-one. M can be expressed as an R-module if and only if R has some representation over it. In particular, every commutative group is a module over the integers, and is either faithful under them or some modular arithmetic.

Another thing to note is that End(M), treated as a group, is also a module over R in a natural way. When R is a field, this constitutes an associative algebra. Modules over fields are called vector spaces.