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 notions are identical if the ring R is commutative.
If R is a field, then a module is also called a vector space. Modules are thus generalizations of vector spaces, and much of the theory of modules consists of recovering desirable properties of vector spaces in the realm or modules over certain rings.
The action of an element r in R is defined to be the map that sends each x to rx (or xr in the case of a right module), and is necessarily a group endomorphism of M. The set of all group endomorphisms of M is denoted End(M) and forms a ring under addition and composition, and the actions of ring elements actually define a ring homomorphism from R to End(M).
Such a homorphism 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 abelian group is a module over the integers, and is either faithful under them or some modular arithmetic.