Module

In abstract algebra, a left R-module consists of some abelian group (M,+) together with a ring of scalars (R,+,*) and an operation R x M -> M (scalar multiplication, usually just written by juxtaposition, i.e. as rx for r in R and x in M) such that

  For all r,s in R, x in M: (rs)x = r(sx)
  For all r,s in R, x in M: (r+s)x = rx+sx
  For all r in R, x,y in M: r(x+y) = rx+ry
  For all 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 an R-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 of modules over certain rings.

Examples

• 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) for n < 0.
• 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

Suppose M is an R-module and N is a subgroup of M. Then N is a submodule (or R-submodule, to be more explicit) if, for any n in N and any r in R, the product rn is in N (or nr for a right module).

If M and N are R-modules, then a map f:M -> N is a homomorphism if, for any m1, m2 in M and r1, r2 in R, f(r1m1 + r2m2) = r1f(m1) + r2f(m2). This, like any homomorphism of mathematical objects, is just a mapping which preserves the structure of the objects.

Alternative definition as representations

If M is a left R-module, then the action of an element r in R is defined to be the map M → M 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_Z(M) and forms a ring under addition and composition, and sending a ring element to its action actually defines a ring homomorphism from R to End_Z(M).

Such a ring homorphism R → End_Z(M) is called a representation of R over the abelian group M; an alternative and equivalent way to defining left R-modules is to say that a left R-module is an abelian group M together with a representation of R over it.

A representation is called faithful if and only if the map R → End_Z(M) is one-to-one. Every abelian group is a module over the integers, and is either faithful under them or some modular arithmetic.