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.
- Every abelian group M is a module over the ring of integers Z if we define nx = x + x + ... + x (n summands) for n ≥ 0, and (-n)x = -(nx).
- If R is any ring and n a natural number, then the cartesian product Rn is a module over R if we use the component-wise operations.
- If M is a smooth manifold, then the smooth functions from M to the real numbers form a ring R. The set of all vector fields defined on M form a module over R, and so do the tensor fields and the differential forms on M.
- The square n-by-n matrices with real entries form a ring R, and the Euclidean space Rn is a left module over this ring if we define the module operation via matrix multiplication.
- If R is any ring and I is any left ideal in R, then I is a left module over R.
Submodules and homomorphisms
- Still missing.
Alternative definition as representations
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.