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 : ${\displaystyle k\in \mathbb {Z} }$ ${\displaystyle k\geq 0}$

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

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 . ${\displaystyle K}$ ${\displaystyle G}$ ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle G}$

See also

• Modulo
• Congruence (number theory)
• Base (module)
• Representation theory
• simple module
• free module
• Group operation
• Modules over main ideal rings
• Module homomorphism
• Sub-module

Web links

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

Alexander von Felbert: Introduction to module theory .

literature

• Siegfried Bosch : Algebra , 7th edition 2009, Springer-Verlag, ISBN 3-540-40388-4 , doi: 10.1007 / 978-3-540-92812-6 .
• LV Kuz'min: modules . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

References and comments

1. ↑ not to be confused with scalar product
2. ↑ ^{a b} David S. Dummit, Richard M. Foote: Abstract Algebra . John Wiley & Sons, Inc., Hoboken, NJ 2004, ISBN 978-0-471-43334-7 .
3. ↑ Here the Abelian group was written additively.
4. ↑ Such a module does not have to have a basis , namely for modules with torsion elements . ${\displaystyle \mathbb {Z} }$
5. ^ Nicolas Bourbaki: Elements of Mathematics, Algebra I, Chapters 1–3 . 2nd Edition. Springer, 1998, ISBN 3-540-64243-9 , § 3. Tensor products , 2., p. 221 ( Internet Archive ).