Module (mathematics) (original) (raw)

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Generalization of vector spaces from fields to rings

In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a (not necessarily commutative) 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.[1]

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

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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 (free modules), the number of elements in a basis need not be the same for all bases (that is to say that they may not have a unique rank) if the underlying ring does not satisfy the invariant basis number condition, unlike vector spaces, which always have a (possibly infinite) basis whose cardinality is then unique. (These last two assertions require the axiom of choice in general, but not in the case of finite-dimensional vector spaces, or certain well-behaved infinite-dimensional vector spaces such as L_p_ spaces.)

Suppose that R is a ring, and 1 is its multiplicative identity. A left _R_-module M consists of an abelian group (M, +) and an operation · : R × M → M such that for all r, s in R and x, y in M, we have

  1. r ⋅ ( x + y ) = r ⋅ x + r ⋅ y {\displaystyle r\cdot (x+y)=r\cdot x+r\cdot y} {\displaystyle r\cdot (x+y)=r\cdot x+r\cdot y},
  2. ( r + s ) ⋅ x = r ⋅ x + s ⋅ x {\displaystyle (r+s)\cdot x=r\cdot x+s\cdot x} {\displaystyle (r+s)\cdot x=r\cdot x+s\cdot x},
  3. ( r s ) ⋅ x = r ⋅ ( s ⋅ x ) {\displaystyle (rs)\cdot x=r\cdot (s\cdot x)} {\displaystyle (rs)\cdot x=r\cdot (s\cdot x)},
  4. 1 ⋅ x = x . {\displaystyle 1\cdot x=x.} {\displaystyle 1\cdot x=x.}

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 R M to emphasize that M is a left _R_-module. A right _R_-module M R is defined similarly in terms of an operation · : M × R → M.

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

( r s ) ⋅ x = s ⋅ ( r ⋅ x ) , {\displaystyle (rs)\cdot x=s\cdot (r\cdot x),} {\displaystyle (rs)\cdot x=s\cdot (r\cdot x),}

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.[2]

An (R,S)-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 (r · x) ∗ s = r ⋅ (x ∗ s) 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.

Submodules and homomorphisms

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Suppose M is a left _R_-module and N is a subgroup of M. Then N is a submodule (or more explicitly an _R_-submodule) if for any n in N and any r in R, the product r ⋅ n (or n ⋅ r for a right _R_-module) is in N.

If X is any subset of an _R_-module M, then the submodule spanned by X is defined to be ⟨ X ⟩ = ⋂ N ⊇ X N {\textstyle \langle X\rangle =\,\bigcap _{N\supseteq X}N} {\textstyle \langle X\rangle =\,\bigcap _{N\supseteq X}N} where N runs over the submodules of M that contain X, or explicitly { ∑ i = 1 k r i x i | r i ∈ R , x i ∈ X } {\textstyle {\bigl \{}\!\sum _{i=1}^{k}r_{i}x_{i}\mathrel {\big |} r_{i}\in R,\,x_{i}\in X{\bigr \}}} {\textstyle {\bigl \{}\!\sum _{i=1}^{k}r_{i}x_{i}\mathrel {\big |} r_{i}\in R,\,x_{i}\in X{\bigr \}}}, which is important in the definition of tensor products of modules.[3]

The set of submodules of a given module M, together with the two binary operations + (the module spanned by the union of the arguments) and ∩, forms a lattice that satisfies the modular law: Given submodules U, _N_1, _N_2 of M such that _N_1 ⊆ _N_2, then the following two submodules are equal: (_N_1 + U) ∩ _N_2 = _N_1 + (U ∩ _N_2).

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 ( r ⋅ m + s ⋅ n ) = r ⋅ f ( m ) + s ⋅ f ( n ) {\displaystyle f(r\cdot m+s\cdot n)=r\cdot f(m)+s\cdot f(n)} {\displaystyle f(r\cdot m+s\cdot n)=r\cdot f(m)+s\cdot f(n)}.

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 f : M → N 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 f : M → N 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(m) for all elements m of M.[4] 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 (see category of modules).

Finitely generated

An _R_-module M is finitely generated if there exist finitely many elements _x_1, ..., x n in M such that every element of M is a linear combination of those elements with coefficients from the ring R.

Cyclic

A module is called a cyclic module if it is generated by one element.

Free

A free _R_-module is a module that has a basis, or equivalently, one that is isomorphic to a direct sum of copies of the 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.

Flat

A module is called flat if taking the tensor product of it with any exact sequence of _R_-modules preserves exactness.

Torsionless

A module is called torsionless if it embeds into its algebraic dual.

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.[5]

Semisimple

A semisimple module is a direct sum (finite or not) of simple modules. Historically these modules are also called completely reducible.

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, but there are indecomposable modules that are not simple (e.g. uniform modules).

Faithful

A faithful module M is one where the action of each r ≠ 0 in R on M is nontrivial (i.e. r ⋅ x ≠ 0 for some x in M). Equivalently, the annihilator of M is the zero ideal.

Torsion-free

A torsion-free module is a module over a ring such that 0 is the only element annihilated by a regular element (non zero-divisor) of the ring, equivalently rm = 0 implies r = 0 or m = 0.

Noetherian

A Noetherian module is a module that satisfies the ascending chain condition on submodules, that is, every increasing chain of submodules becomes stationary after finitely many steps. Equivalently, every submodule is finitely generated.

Artinian

An Artinian module is a module that satisfies the descending chain condition on submodules, that is, every decreasing chain of submodules becomes stationary after finitely many steps.

Graded

A graded module is a module with a decomposition as a direct sum M = ⨁x M x over a graded ring R = ⨁x R x such that R x M y ⊆ M x+y for all x and y.

Uniform

A uniform module is a module in which all pairs of nonzero submodules have nonzero intersection.

Relation to representation theory

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A representation of a group G over a field k is a module over the group ring _k_[_G_].

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 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 the abelian group M over the ring R; 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 M over R. Such a representation R → EndZ(M) may also be called a ring action of R on M.

A representation is called faithful 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 the ring of integers modulo n, Z/nZ, for some n.

A ring R corresponds to a preadditive category R with a single object. With this understanding, a left _R_-module is just a covariant additive functor from R to the category Ab of abelian groups, and 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 (see sheaf of modules). These form a category O_X_-Mod, and play an important role in modern algebraic geometry. If X has only a single point, then this is a module category in the old sense over the commutative ring O_X_(X).

One can also consider modules over a semiring. Modules over rings are abelian groups, but modules over semirings are only commutative monoids. Most applications of modules are still possible. In particular, for any semiring S, the matrices over S form a semiring over which the tuples of elements from S are a module (in this generalized sense only). This allows a further generalization of the concept of vector space incorporating the semirings from theoretical computer science.

Over near-rings, one can consider near-ring modules, a nonabelian generalization of modules.[_citation needed_]

  1. ^ Hungerford (1974) Algebra, Springer, p 169: "Modules over a ring are a generalization of abelian groups (which are modules over Z)."
  2. ^ Dummit, David S. & Foote, Richard M. (2004). Abstract Algebra. Hoboken, NJ: John Wiley & Sons, Inc. ISBN 978-0-471-43334-7.
  3. ^ Mcgerty, Kevin (2016). "ALGEBRA II: RINGS AND MODULES" (PDF).
  4. ^ Ash, Robert. "Module Fundamentals" (PDF). Abstract Algebra: The Basic Graduate Year.
  5. ^ Jacobson (1964), p. 4, Def. 1