Module (mathematics)

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In abstract algebra, the notion of a module over a ring is the common generalizations of two of the most important notions in algebra, vector space, and abelian group.

Motivation

In a vector space, the set of scalars forms a field and acts on the vectors by scalar multiplication, subject to certain formal laws such as the distributive law. In a module, the scalars need only be a ring, so the module concept represents a significant generalization.

Much of the theory of modules consists of extending as many as possible of the desirable properties of vector spaces to the realm of modules over a "well-behaved" ring, such as a principal ideal domain. However, modules can be quite a bit more complicated than vector spaces; for instance, not all modules have a basis, and even those that do, free modules, behave significantly differently from vector spaces in some respects.

Modules form the core notion of commutative algebra, which is essential in many important fields of mathematics, including

Definition

Specifically, a left R-module over the ring R consists of an abelian group (M, +) and an operation R × MM (called 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. r(x+y) = rx+ry
  2. (r+s)x = rx+sx
  3. (rs)x = r(sx)
  4. 1x = x

Usually, we simply write "a left R-module M" or RM. A right R-module M or MR is defined similarly, only the ring acts on the right, i.e. we have a scalar multiplication of the form M × RM, and the above axioms are written with scalars r and s on the right of x and y.

Authors who do not require rings to be unital omit condition 4 in the above definition, and call the above structures "unital left modules". In this article however, all rings and modules are assumed to be unital.

A bimodule is a module which is both a left module and a right module.

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.
  • The concept of a Z-module agrees with the notion of an abelian group. That is, every abelian group is a module over the ring of integers Z in a unique way. For n > 0, let nx = x + x + ... + x (n summands), 0x = 0, and (−n)x = −(nx).
  • 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. Hence when n=1, R is an R-module, where the scalar multiplication is just ring multiplication. The case n=0 yields the trivial R-module {0} consisting only of its identity element. Modules of this type are called free and the number n is then the rank of the free module.
  • If S is a nonempty set, M is a left R-module, and MS is the collection of all functions f : SM, then with addition and scalar multiplication in MS defined by (f + g)(s) = f(s) + g(s) and (rf)(s) = rf(s), MS is a left R-module. The right R-module case is analogous. In particular, if R is commutative then the collection of R-module homomorphisms h : MN (see below) is an R-module (and in fact a submodule of NM).
  • If X is a smooth manifold, then the smooth functions from X to the real numbers form a ring C(X). The set of all smooth vector fields defined on X form a module over C(X), 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 : MN 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 : MN 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,...,xn 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 ≠ 0 in R on M is nontrivial (i.e. rx ≠ 0 for some x in 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.

Graded. A graded module can be decomposed as a free product of modules.

Relation to representation theory

If M is a left R-module, then the action of an element r in R is defined to be the map MM 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 EndZ(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 EndZ(M).

Such a ring homomorphism R → EndZ(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 → EndZ(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, OX) and consider the sheaves of OX-modules. These form a category OX-Mod. If X has only a single point, then this is a module category in the old sense over the commutative ring OX(X).

See also

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, ISBN 0387978453, ISBN 3540978453