Wikipedia 10K Redux by Reagle from Starling archive. Bugs abound!!!

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A left R-module consists of some commutative group (M,+) together with a RinG of scalars (R,+,*), together with an operation RxM->M (scalar multiplication, usually just denoted *) such that 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 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 EndoMorphism''s'' 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 under the IntegerNumbers, and is either faithful under them or some ModularArithmetic. A module under a field is called a VectorSpace.