# Module (mathematics)

A module [ moːdʊl ] (masculine, plural modules [ moːdʊln ], the declination is similar to that of consul ; of Latin modulus , diminutive of mode , "degree", "unit") is an algebraic structure that a generalization of a vector space represents.

Similar to rings , a module is understood to mean something different depending on the sub-area and textbook. The definitions of morphisms as well as substructures and superstructures are also slightly different. In mathematical terms, these different module terms are different categories .

## Modules over a commutative ring with a single element

A module over a commutative ring or module for short is an additive Abelian group together with a mapping ${\ displaystyle (R, +, \ cdot)}$${\ displaystyle R}$ ${\ displaystyle (M, +)}$

${\ displaystyle R \ times M \ to M, \ quad (r, m) \ mapsto r \ cdot m}$           (called multiplication by scalars, scalar multiplication),

so that:

${\ displaystyle r_ {1} \ cdot (r_ {2} \ cdot m) = (r_ {1} \ cdot r_ {2}) \ cdot m}$
${\ displaystyle (r_ {1} + r_ {2}) \ cdot m = r_ {1} \ cdot m + r_ {2} \ cdot m}$
${\ displaystyle r \ cdot (m_ {1} + m_ {2}) = r \ cdot m_ {1} + r \ cdot m_ {2}}$

If one requires additionally for an identity with ${\ displaystyle (R, +, \ cdot)}$${\ displaystyle 1}$

${\ displaystyle 1 \ cdot m = m}$,

so one calls the module unitary (English: unital ). Some authors generally require the existence of a unit element for rings, and then likewise for modules over rings. If there is a body , i.e. also forms an Abelian group, then the unitary modules are over just the vector spaces over . ${\ displaystyle R}$${\ displaystyle R}$${\ displaystyle (R \ backslash \ {0_ {R} \}, \, \ cdot)}$${\ displaystyle R}$${\ displaystyle R}$

Comment: The concept of vector space is actually superfluous, since it is a special case of the more general concept of unitary module. In fact, however, the additional condition that there is a body enables so many results that are not correct in the general situation that it is customary to separate the special case from the general case with a separate term. ${\ displaystyle R}$

The study of modules on commutative rings is the subject of commutative algebra .

### Abelian groups

Each additive Abelian group is uniquely a unitary module; H. a unitary module over the commutative ring of integers. Be . Because of ${\ displaystyle G}$${\ displaystyle \ mathbb {Z}}$${\ displaystyle m \ in G}$

${\ displaystyle 1 \ cdot m = m, \, 0 \ cdot m = {\ mathfrak {0}}}$

must apply to with : ${\ displaystyle k \ in \ mathbb {Z}}$${\ displaystyle k \ geq 0}$

${\ displaystyle k \ cdot m = \ underbrace {(1+ \ dotsb +1)} _ {k {\ text {times}}} \ cdot m = \ underbrace {m + \ dotsb + m} _ {k {\ text {times}}}}$

and analog:

${\ displaystyle (-k) \ cdot m = - \ underbrace {(m + \ dotsb + m)} _ {k {\ text {-mal}}}}$

Since this only possible connection fulfills the modulus axioms, the claim follows. The following number ranges are additive groups and thus modules: ${\ displaystyle \ mathbb {Z}}$

• the whole numbers themselves${\ displaystyle \ mathbb {Z}}$
• the rational numbers ${\ displaystyle \ mathbb {Q}}$
• the real numbers ${\ displaystyle \ mathbb {R}}$
• the algebraic numbers resp.${\ displaystyle \ mathbb {A}}$${\ displaystyle \ mathbb {A} \ cap \ mathbb {R}}$
• the complex numbers ${\ displaystyle \ mathbb {C}}$

### Upper rings as modules

Is a top ring of , it is by definition an Abelian group. ${\ displaystyle (S, +, \ cdot)}$${\ displaystyle (R, +, \ cdot)}$${\ displaystyle (S, +)}$

If the ring multiplication of is restricted to the set , then this defines the necessary scalar multiplication in order to consider it as a module over in a natural way . If and if they have the same single element, the module is unitary. ${\ displaystyle S}$${\ displaystyle R \ times S}$${\ displaystyle S}$${\ displaystyle R}$${\ displaystyle R}$${\ displaystyle S}$

If and even bodies are, one speaks in this situation of a body expansion . The module structure then becomes a vector space structure as described above. The consideration of this vector space structure is an indispensable aid in the investigation of body extensions. ${\ displaystyle R}$${\ displaystyle S}$

Note: The number ranges mentioned in the previous chapter are all upper rings of , which also shows that they are naturally modules. ${\ displaystyle \ mathbb {Z}}$${\ displaystyle \ mathbb {Z}}$

### Vector spaces with a linear map in themselves

Let be the polynomial ring over a body . Then, the corresponding -modules one-to-one pairs consisting of a vector space and an endomorphism on . ${\ displaystyle K [X]}$ ${\ displaystyle K}$${\ displaystyle K [X]}$${\ displaystyle (V, A)}$${\ displaystyle K}$${\ displaystyle V}$ ${\ displaystyle A}$${\ displaystyle V}$

• Be a module. We note that a vector space is as in embedded. Let this vector space. The pair that belongs to it is now , being through${\ displaystyle M}$${\ displaystyle K [X]}$${\ displaystyle M}$${\ displaystyle K}$${\ displaystyle K}$${\ displaystyle K [X]}$${\ displaystyle V}$${\ displaystyle M}$${\ displaystyle (V, A)}$${\ displaystyle A}$
${\ displaystyle V \ to V, \ quad v \ mapsto X \ cdot v.}$
given is.
• We define a module structure for a pair${\ displaystyle (V, A)}$${\ displaystyle K [X]}$
${\ displaystyle X \ cdot v: = A (v)}$
and continue the -linear on , d. H. for all ${\ displaystyle K}$${\ displaystyle K [X]}$
${\ displaystyle p (X) = a_ {0} + a_ {1} X + a_ {2} X ^ {2} + \ dotsb + a_ {n} X ^ {n} \ in K [X]}$
let's set
${\ displaystyle p (X) \ cdot v: = (p (A)) (v): = a_ {0} \ cdot v + a_ {1} \ cdot A (v) + a_ {2} \ cdot A ^ {2} (v) + \ dotsb + a_ {n} \ cdot A ^ {n} (v)}$.

### Ring ideals

Each ring is a module on itself with ring multiplication as an operation. The sub-modules then correspond exactly to the ideals of (since in this section is commutative, we do not need to distinguish between left and right ideals). ${\ displaystyle R}$${\ displaystyle R}$

## Modules over any ring

Let it be a ring . If this ring is not (necessarily) commutative, a distinction must be made between left and right modules. ${\ displaystyle (R, +, \ cdot)}$

A - left module is an Abelian group along with a ring and a map ${\ displaystyle R}$ ${\ displaystyle (M, +)}$${\ displaystyle (R, +, \ cdot)}$

${\ displaystyle R \ times M \ to M, \ quad (r, m) \ mapsto r \ cdot m = rm,}$

which is additive in both arguments, d. H. for all true ${\ displaystyle r, r_ {1}, r_ {2} \ in R, \, m, m_ {1}, m_ {2} \ in M}$

• ${\ displaystyle (r_ {1} + r_ {2}) \ cdot m = r_ {1} \ cdot m + r_ {2} \ cdot m}$ and
• ${\ displaystyle r \ cdot (m_ {1} + m_ {2}) = r \ cdot m_ {1} + r \ cdot m_ {2},}$

and for the

• ${\ displaystyle r_ {1} \ cdot (r_ {2} \ cdot m) = (r_ {1} \ cdot r_ {2}) \ cdot m}$ for all ${\ displaystyle r_ {1}, r_ {2} \ in R, \ m \ in M}$

applies.

If it is assumed that there is a unitary ring with a one-element , then it is usually also required that the left module is unitary (English: unital ), i.e. H. ${\ displaystyle (R, +, \ cdot)}$${\ displaystyle 1}$${\ displaystyle R}$

• ${\ displaystyle 1 \ cdot m = m}$for everyone .${\ displaystyle m \ in M}$

Some authors generally require the existence of a single element for rings and modules.

A right module is defined similarly, except that the scalars of the ring act from the right on the elements of : A - right module is an Abelian group together with a mapping that is additive in both arguments ${\ displaystyle M}$
${\ displaystyle R}$${\ displaystyle M}$

${\ displaystyle M \ times R \ to M, \ quad (m, r) \ mapsto m \ cdot r = mr,}$

so that

${\ displaystyle (m \ cdot r_ {1}) \ cdot r_ {2} = m \ cdot (r_ {1} \ cdot r_ {2})}$ for all ${\ displaystyle r_ {1}, r_ {2} \ in R, \ m \ in M.}$

A right module over a unitary ring with one element is unitary if ${\ displaystyle 1}$

${\ displaystyle m \ cdot 1 = m}$applies to all .${\ displaystyle m \ in M}$

If commutative , the terms left and right module match (except for the spelling), and one speaks simply of - modules . Usually the above notation is used for left modules. ${\ displaystyle R}$ ${\ displaystyle R}$

### Alternative definitions

• A left module is an Abelian group together with a (possibly unitary) ring homomorphism${\ displaystyle R}$${\ displaystyle M}$
${\ displaystyle R \ to \ mathrm {End} _ {\ mathbb {Z}} (M).}$
It is the ring of endomorphism of the concatenation as a product: ${\ displaystyle \ mathrm {End} _ {\ mathbb {Z}} (M)}$${\ displaystyle M}$
${\ displaystyle (f_ {1} \ cdot f_ {2}) (m): = f_ {1} (f_ {2} (m))}$ For ${\ displaystyle f_ {1}, f_ {2} \ in \ mathrm {End} _ {\ mathbb {Z}} (M), m \ in M.}$
• A right module is an Abelian group together with a (possibly unitary) ring homomorphism${\ displaystyle R}$${\ displaystyle M}$
${\ displaystyle R \ to (\ mathrm {End} _ {\ mathbb {Z}} (M)) ^ {\ mathrm {op}}.}$
It should be the counter-ring of endomorphism ring, that is the ring of endomorphisms of the legal chaining as a product: ${\ displaystyle (\ mathrm {End} _ {\ mathbb {Z}} (M)) ^ {\ mathrm {op}}}$${\ displaystyle M}$
${\ displaystyle (f_ {1} \ cdot f_ {2}) (m): = f_ {2} (f_ {1} (m))}$ For ${\ displaystyle f_ {1}, f_ {2} \ in (\ mathrm {End} _ {\ mathbb {Z}} (M)) ^ {\ mathrm {op}}, m \ in M.}$

### Bimodules

Let it be and rings. Then a - -Bimodule is an Abelian group together with a -Left module- and a -Right module structure, so that ${\ displaystyle R}$${\ displaystyle S}$${\ displaystyle R}$${\ displaystyle S}$${\ displaystyle M}$${\ displaystyle R}$${\ displaystyle S}$

${\ displaystyle (r \ cdot m) \ cdot s = r \ cdot (m \ cdot s)}$ For ${\ displaystyle r \ in R, m \ in M, s \ in S}$

applies.

For unitary rings and a unitary - -Bimodule (i.e. with for all ) can alternatively be described as an Abelian group together with a unitary ring homomorphism ${\ displaystyle R}$${\ displaystyle S}$${\ displaystyle R}$${\ displaystyle S}$${\ displaystyle 1_ {R} \ cdot m = m \ cdot 1_ {S} = m}$${\ displaystyle m \ in M}$${\ displaystyle M}$

${\ displaystyle R \ otimes _ {\ mathbb {Z}} S ^ {\ mathrm {op}} \ to \ mathrm {End} _ {\ mathbb {Z}} \, M.}$

That means: A unitary - -Bimodule is nothing more than a unitary -Link module. ${\ displaystyle R}$${\ displaystyle S}$${\ displaystyle R \ otimes _ {\ mathbb {Z}} S ^ {\ mathrm {op}}}$

## Change of ring

${\ displaystyle R}$and be rings and be a ring homomorphism. The regulation defines for each module${\ displaystyle S}$${\ displaystyle \ rho \ colon S \ to R}$${\ displaystyle R}$${\ displaystyle M}$

${\ displaystyle (s, m) \ mapsto \ rho (s) m}$

a module structure called the one associated with and the module structure . This module is labeled with or with . If in particular a subring of and is the canonical embedding, then the module obtained by restricting the scalars from to is called. ${\ displaystyle S}$${\ displaystyle M}$${\ displaystyle \ rho}$${\ displaystyle R}$${\ displaystyle S}$${\ displaystyle \ rho _ {*} (M)}$${\ displaystyle M _ {[S]}}$${\ displaystyle S}$${\ displaystyle R}$${\ displaystyle \ rho}$${\ displaystyle \ rho _ {*} (M)}$${\ displaystyle R}$${\ displaystyle S}$${\ displaystyle S}$

If a sub-module is of , then is a sub-module of and${\ displaystyle N}$${\ displaystyle M}$${\ displaystyle \ rho _ {*} (N)}$${\ displaystyle \ rho _ {*} (M)}$${\ displaystyle \ rho _ {*} (M / N) = \ rho _ {*} (M) / \ rho _ {*} (N).}$

## Modules over an associative algebra

If a commutative ring and an associative R -algebra , then a - left module is a module together with a module homomorphism ${\ displaystyle R}$${\ displaystyle A}$${\ displaystyle A}$${\ displaystyle R}$${\ displaystyle M}$${\ displaystyle R}$

${\ displaystyle A \ otimes _ {R} M \ to M, \ quad a \ otimes m \ mapsto am,}$

so that

${\ displaystyle a_ {1} (a_ {2} m) = (a_ {1} a_ {2}) m}$ For ${\ displaystyle a_ {1}, a_ {2} \ in A, m \ in M}$

applies.

A - Law Module is a module together with a -Modulhomomorphismus ${\ displaystyle A}$${\ displaystyle R}$${\ displaystyle M}$${\ displaystyle R}$

${\ displaystyle M \ otimes _ {R} A \ to M, \ quad m \ otimes a \ mapsto ma,}$

so that

${\ displaystyle (ma_ {1}) a_ {2} = m (a_ {1} a_ {2})}$ For ${\ displaystyle a_ {1}, a_ {2} \ in A, m \ in M}$

applies.

Unitary modules and bimodules are defined analogously to the case of rings.

## Modules over a Lie algebra

Let it be a Lie algebra over a field . A - module or a representation of a vector space together with a -bilinearen Figure ${\ displaystyle {\ mathfrak {g}}}$ ${\ displaystyle K}$${\ displaystyle {\ mathfrak {g}}}$${\ displaystyle {\ mathfrak {g}}}$${\ displaystyle K}$${\ displaystyle M}$${\ displaystyle K}$

${\ displaystyle {\ mathfrak {g}} \ times M \ to M, \; (X, m) \ mapsto X \ cdot m,}$

so that

${\ displaystyle [X, Y] \ cdot m = X \ cdot (Y \ cdot m) -Y \ cdot (X \ cdot m)}$ For ${\ displaystyle X, Y \ in {\ mathfrak {g}}, m \ in M}$

applies.

Alternatively, a module is a vector space along with a homomorphism of Lie algebras over${\ displaystyle {\ mathfrak {g}}}$${\ displaystyle K}$${\ displaystyle M}$${\ displaystyle K}$

${\ displaystyle {\ mathfrak {g}} \ to {\ mathfrak {gl}} (M);}$

where is the -algebra of the endomorphisms of with the commutator as a Lie bracket. ${\ displaystyle {\ mathfrak {gl}} (M)}$${\ displaystyle K}$${\ displaystyle M}$

${\ displaystyle {\ mathfrak {g}}}$-Modules are the same as modules under the universal enveloping algebra of . ${\ displaystyle {\ mathfrak {g}}}$

## Modules over a group

It is a group . A module or, more precisely, left module is an Abelian group together with an outer two-digit link${\ displaystyle (G, *)}$${\ displaystyle G}$${\ displaystyle G}$${\ displaystyle (M, +)}$

${\ displaystyle G \ times M \ to M, \; (g, m) \ mapsto g \ cdot m}$,

so that

${\ displaystyle g \ cdot (m_ {1} + m_ {2}) = g \ cdot m_ {1} + g \ cdot m_ {2}}$ for all ${\ displaystyle g \ in G, m_ {1}, m_ {2} \ in M}$

and

${\ displaystyle (g_ {1} * g_ {2}) \ cdot m = g_ {1} \ cdot (g_ {2} \ cdot m)}$ for all ${\ displaystyle g_ {1}, g_ {2} \ in G, m \ in M}$

such as

${\ displaystyle e \ cdot m = m}$for the neutral element of and for everyone${\ displaystyle e}$${\ displaystyle G}$${\ displaystyle m \ in M}$

applies.

A - law module is defined similarly; the second condition is through ${\ displaystyle G}$

${\ displaystyle m \ cdot (g_ {1} * g_ {2}) = (m \ cdot g_ {1}) \ cdot g_ {2}}$ for all ${\ displaystyle g_ {1}, g_ {2} \ in G, m \ in M}$

to replace.

Alternatively, a - (left) module is an Abelian group together with a group homomorphism ${\ displaystyle G}$${\ displaystyle (M, +)}$

${\ displaystyle G \ to \ mathrm {Aut} _ {\ mathbb {Z}} (M),}$

here is the group of automorphisms from with the link ${\ displaystyle \ mathrm {Aut} _ {\ mathbb {Z}} (M) = (\ mathrm {End} _ {\ mathbb {Z}} (M)) ^ {\ times}}$${\ displaystyle M}$

${\ displaystyle (f_ {1} \ circ f_ {2}) (m) = f_ {1} (f_ {2} (m))}$ For ${\ displaystyle f_ {1}, f_ {2} \ in \ mathrm {Aut} _ {\ mathbb {Z}} (M), m \ in M.}$

A right module is an Abelian group together with a group homomorphism ${\ displaystyle G}$${\ displaystyle (M, +)}$

${\ displaystyle G \ to (\ mathrm {Aut} _ {\ mathbb {Z}} (M)) ^ {\ mathrm {op}},}$

the product on is through ${\ displaystyle (\ mathrm {Aut} _ {\ mathbb {Z}} (M)) ^ {\ mathrm {op}}}$

${\ displaystyle (f_ {1} \ bullet f_ {2}) (m): = f_ {2} (f_ {1} (m))}$ For ${\ displaystyle f_ {1}, f_ {2} \ in (\ mathrm {Aut} _ {\ mathbb {Z}} (M)) ^ {\ mathrm {op}}, m \ in M}$

given.

If further is a ring, then a - module is an Abelian group with a module structure and a module structure which are compatible in the following sense: ${\ displaystyle R}$${\ displaystyle G}$${\ displaystyle R}$${\ displaystyle R}$${\ displaystyle G}$

${\ displaystyle r \ cdot (g \ cdot m) = g \ cdot (r \ cdot m)}$ For ${\ displaystyle r \ in R, g \ in G, m \ in M.}$

Alternatively, a - module is a module together with a group homomorphism ${\ displaystyle G}$${\ displaystyle R}$${\ displaystyle R}$

${\ displaystyle G \ to \ mathrm {Aut} _ {R} (M),}$

this is the group of automorphisms of the module. ${\ displaystyle G \ to \ mathrm {Aut} _ {R} (M)}$${\ displaystyle M}$${\ displaystyle R}$

${\ displaystyle G}$- modules are the same as modules above the group ring . ${\ displaystyle R}$ ${\ displaystyle R [G]}$

If a body is special , the concept of the - module corresponds to that of the linear representation of . ${\ displaystyle K}$${\ displaystyle G}$${\ displaystyle K}$${\ displaystyle K}$${\ displaystyle G}$

Wiktionary: Module  - explanations of meanings, word origins, synonyms, translations

Alexander von Felbert: Introduction to module theory .

## literature

4. Such a module does not have to have a basis , namely for modules with torsion elements .${\ displaystyle \ mathbb {Z}}$