Module (mathematics)


In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of a module also generalizes the notion of an abelian group, since the abelian groups are exactly the modules over the ring of integers.
Like a vector space, a module is an additive abelian group, and scalar multiplication is distributive over the operations of addition between elements of the ring or module and is compatible with the ring multiplication.
Modules are very closely related to the representation theory of groups. They are also one of the central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology.

Introduction and definition

Motivation

In a vector space, the set of scalars is a field and acts on the vectors by scalar multiplication, subject to certain axioms such as the distributive law. In a module, the scalars need only be a ring, so the module concept represents a significant generalization. In commutative algebra, both ideals and quotient rings are modules, so that many arguments about ideals or quotient rings can be combined into a single argument about modules. In non-commutative algebra, the distinction between left ideals, ideals, and modules becomes more pronounced, though some ring-theoretic conditions can be expressed either about left ideals or left modules.
Much of the theory of modules consists of extending as many of the desirable properties of vector spaces as possible 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 for those that do, the number of elements in a basis need not be the same for all bases if the underlying ring does not satisfy the invariant basis number condition, unlike vector spaces, which always have a basis whose cardinality is then unique.

Formal definition

Suppose that R is a ring, and 1 is its multiplicative identity.
A left R-module M consists of an abelian group and an operation such that for all r, s in R and x, y in M, we have
The operation · is called scalar multiplication. Often the symbol · is omitted, but in this article we use it and reserve juxtaposition for multiplication in R. One may write RM to emphasize that M is a left R-module. A right R-module MR is defined similarly in terms of an operation.
The qualificative of left- or right-module does not depend on whether the scalars are written on the left or on the right, but on the property 3: if, in the above definition, the property 3 is replaced by
one gets a right-module, even if the scalars are written on the left. However, writing the scalars on the left for left-modules and on the right for right modules makes the manipulation of property 3 much easier.
Authors who do not require rings to be unital omit condition 4 in the definition above; they would call the structures defined above "unital left R-modules". In this article, consistent with the glossary of ring theory, all rings and modules are assumed to be unital.
An -bimodule is an abelian group together with both a left scalar multiplication · by elements of R and a right scalar multiplication ∗ by elements of S, making it simultaneously a left R-module and a right S-module, satisfying the additional condition for all r in R, x in M, and s in S.
If R is commutative, then left R-modules are the same as right R-modules and are simply called R-modules. Most often the scalars are written on the left in this case.

Examples

  • If K is a field, then K-modules are called K-vector spaces.
  • If K is a field, and K a univariate polynomial ring, then a Polynomial ring#Modules|K-module M is a K-module with an additional action of x on M by a group homomorphism that commutes with the action of K on M. In other words, a K-module is a K-vector space M combined with a linear map from M to M. Applying the structure theorem for finitely generated modules over a principal ideal domain to this example shows the existence of the rational and Jordan canonical forms.
  • 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, let ,, and. Such a module need not have a basis—groups containing torsion elements do not.
  • The decimal fractions form a module over the integers. Only singletons are linearly independent sets, but there is no singleton that can serve as a basis, so the module has no basis and no rank, in the usual sense of linear algebra. However this module has a torsion-free rank equal to 1.
  • If R is any ring and n a natural number, then the cartesian product Rn is both a left and right R-module over R if we use the component-wise operations. Hence when, R is an R-module, where the scalar multiplication is just ring multiplication. The case yields the trivial R-module consisting only of its identity element. Modules of this type are called free and if R has invariant basis number the number n is then the rank of the free module.
  • If Mn is the ring of matrices over a ring R, M is an Mn-module, and ei is the matrix with 1 in the -entry, then eiM is an R-module, since. So M breaks up as the direct sum of R-modules,. Conversely, given an R-module M0, then M0⊕n is an Mn-module. In fact, the category of R-modules and the category of Mn-modules are equivalent. The special case is that the module M is just R as a module over itself, then Rn is an Mn-module.
  • If S is a nonempty set, M is a left R-module, and MS is the collection of all functions, then with addition and scalar multiplication in MS defined pointwise by and, 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 is an R-module.
  • If X is a smooth manifold, then the smooth functions from X to the real numbers form a ring C∞. The set of all smooth vector fields defined on X forms a module over C∞, and so do the tensor fields and the differential forms on X. More generally, the sections of any vector bundle form a projective module over C∞, and by Swan's theorem, every projective module is isomorphic to the module of sections of some vector bundle; the category of C∞-modules and the category of vector bundles over X are equivalent.
  • If R is any ring and I is any left ideal in R, then I is a left R-module, and analogously right ideals in R are right R-modules.
  • If R is a ring, we can define the opposite ring Rop, which has the same underlying set and the same addition operation, but the opposite multiplication: if in R, then in Rop. Any left ''R-module M'' can then be seen to be a right module over Rop, and any right module over R can be considered a left module over Rop.
  • Modules over a Lie algebra are modules over its universal enveloping algebra.
  • If R and S are rings with a ring homomorphism, then every S-module M is an R-module by defining. In particular, S itself is such an R-module.

    Submodules and homomorphisms

Suppose M is a left R-module and N is a subgroup of M. Then N is a submodule if for any n in N and any r in R, the product is in N.
If X is any subset of an R-module M, then the submodule spanned by X is defined to be where N runs over the submodules of M that contain X, or explicitly, which is important in the definition of tensor products of modules.
The set of submodules of a given module M, together with the two binary operations + and ∩, forms a lattice that satisfies the modular law:
Given submodules U, N1, N2 of M such that, then the following two submodules are equal:.
If M and N are left R-modules, then a map is a homomorphism of R-modules if for any m, n in M and r, s in R,
This, like any homomorphism of mathematical objects, is just a mapping that preserves the structure of the objects. Another name for a homomorphism of R-modules is an R-linear map.
A bijective module homomorphism is called a module isomorphism, and the two modules M and N 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 is the submodule of M consisting of all elements that are sent to zero by f, and the image of f is the submodule of N consisting of values f for all elements m of M. The isomorphism theorems familiar from groups and vector spaces are also valid for R-modules.
Given a ring R, the set of all left R-modules together with their module homomorphisms forms an abelian category, denoted by R-Mod.