Module (mathematics)

In abstract algebra, modules are generalizations of vector spaces to arbitrary ring of scalars. Much of the theory of modules consists of recovering desirable properties of vector spaces in the realm of modules over certain rings. However, modules can be quite a bit more complicated than vector spaces; for instance, not all modules have a basis.

Definition

Specifically, a left module over the ring R consists of an abelian group (M, +) and an operation R × 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,y in M, we have

1. (rs)x = r(sx)
2. (r+s)x = rx+sx
3. r(x+y) = rx+ry
4. 1x = x

Usually, we simply write "a left R-module M" or _RM.

Some authors omit condition 4 for the general definition of left modules, and call the above defined structures "unital left modules". In this encyclopedia however, all modules are assumed to be unital, and all rings are assumed to have a one.

A right R-module M or M_R is defined similarly, only the ring acts on the right, i.e. we have a scalar multiplication of the form M × R -> M, and the above three axioms are written with scalars r and s on the right of x and y.

If R is commutative, then left R-modules are the same as right R-modules and are simply called R-modules.

Examples

• If K is a field, then the concepts "K-vector space" and K-module are identical.
• 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, 0x = 0, and (-n)x = -(nx) for n < 0.
• If R is any ring and n a natural number, then the cartesian product Rⁿ is both a left and a right module over R if we use the component-wise operations. The case n=0 yields the trivial R-module {0} consisting only of its identity element.
• If X is a smooth manifold, then the smooth functions from X to the real numbers form a ring R. The set of all smooth vector fields defined on X form a module over R, and so do the tensor fields and the differential forms on X.
• 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. Analogously of course, right ideals are right modules.

Submodules and homomorphisms

Suppose M is a left 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 left R-modules, then a map f : M -> N is a homomorphism of R-modules if, for any m, n in M and r, s in R,

f(rm + sn) = rf(m) + sf(n).

This, like any homomorphism of mathematical objects, is just a mapping which preserves the structure of the objects.

A bijective module homomorphism is an isomorphism of modules, and the two modules are called isomorphic. Two isomorphic modules are identical for all practical purposes, differing solely in the notation for their elements.

The kernel of a module homomorphism f : M -> N is the submodule of M consisting of all elements that are sent to zero by f. The isomorphism theorems familiar from abelian groups and vector spaces are also valid for R-modules.

The left R-modules, together with their module homomorphisms, form a category, written as R-Mod. This is an abelian category.

Types of modules

Finitely generated. A module M is finitely generated if there exist finitely many elements x₁,...,x_n in M such that every element of M is a linear combination of those elements with coefficients from the scalar ring R.

Free. A free module is a module that has a basis, or equivalently, one that is isomorphic to a direct sum of copies of the scalar ring R. These are the modules that behave very much like vector spaces.

Projective. Projective modules are direct summands of free modules and share many of their desirable properties.

Injective. Injective modules are defined dually to projective modules.

Simple. A simple module S is a module that is not {0} and whose only submodules are {0} and S. Simple modules are sometimes called irreducible.

Indecomposable. An indecomposable module is a non-zero module that cannot be written as a direct sum of two non-zero submodules. Every simple module is indecomposable.

Faithful. A faithful module M is one where the action of each r in R gives an injective map M→M. Equivalently, the annihilator of M is the zero ideal.

Noetherian. A noetherian module is a module whose every submodule is finitely generated. Equivalently, every increasing chain of submodules becomes stationary after finitely many steps.

Artinian. An artinian module is a module in which every decreasing chain of submodules becomes stationary after finitely many steps.

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 the abelian group (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 r of R 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 of 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 injective. In terms of modules, this means that if r is an element of R such that rx=0 for all x in M, then r=0. Every abelian group is a faithful module over the integers or over some modular arithmetic Z/nZ.

Generalizations

Any ring R can be viewed as a preadditive category with a single object. With this understanding, a left R-module is nothing but a (covariant) additive functor from R to the category Ab of abelian groups. Right R-modules are contravariant additive functors. This suggests that, if C is any preadditive category, a covariant additive functor from C to Ab should be considered a generalized left module over C; these functors form a functor category C-Mod which is the natural generalization of the module category R-Mod.

Modules over commutative rings can be generalized in a different direction: take a ringed space (X, O_X) and consider the sheaves of O_X-modules. These form a category O_X-Mod. If X has only a single point, then this is a module category in the old sense over the commutative ring O_X(X).

References

• F.W. Anderson and K.R. Fuller: Rings and Categories of Modules, Graduate Texts in Mathematics, Vol. 13, 2 nd Ed., Springer-Verlag, New York, 1992