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#REDIRECT [[Module]]
A left R-module consists of some commutative [[Mathematical Group|group]] (M,+) together with a [[Mathematical ring|ring]] of scalars (R,+,*), together with an operation R x M->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 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 [[integer]]s, 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 space]]s.
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