Free algebra
Encyclopedia
In mathematics
, especially in the area of abstract algebra
known as ring theory
, a free algebra is the noncommutative analogue of a polynomial ring
(which may be regarded as a free commutative algebra).
, the free (associative, unital) algebra
on n indeterminate
s {X1,...,Xn} is the free R-module
with a basis consisting of all words over the alphabet {X1,...,Xn} (including the empty word, which is the unity of the free algebra). This R-module becomes an R-algebra
by defining a multiplication as follows: the product of two basis elements is the concatenation
of the corresponding words:
and the product of two arbitrary elements is thus uniquely determined (because the multiplication in an R-algebra must be R-bilinear). This R-algebra is denoted R⟨X1,...,Xn⟩. This construction can easily be generalized to an arbitrary set X of indeterminates.
In short, for an arbitrary set
, the free (associative, unital) R-algebra
on X is
with the R-bilinear multiplication that is concatenation on words, where X* denotes the free monoid on X (i.e. words on the letters Xi),
denotes the external direct sum
, and Rw denotes the free R-module
on 1 element, the word w.
For example, in R⟨X1,X2,X3,X4⟩, for scalars α,β,γ,δ ∈R, a concrete example of a product of two elements is
.
where
are elements of R and all but finitely many of these elements are zero. This explains why the elements of R⟨X1,...,Xn⟩ are often denoted as "non-commutative polynomials" in the "variables" (or "indeterminates") X1,...,Xn; the elements 
are said to be "coefficients" of these polynomials, and the R-algebra R⟨X1,...,Xn⟩ is called the "non-commutative polynomial algebra over R in n indeterminates". Note that unlike in an actual polynomial ring, the variables do not commute. For example X1X2 does not equal X2X1.
More generally, one can construct the free algebra R⟨E⟩ on any set E of generators
. Since rings may be regarded as Z-algebras, a free ring on E can be defined as the free algebra Z'E⟩.
Over a field
, the free algebra on n indeterminates can be constructed as the tensor algebra
on an n-dimensional vector space
. For a more general coefficient ring, the same construction works if we take the free module
on n generators
.
The construction of the free algebra on E is functor
ial in nature and satisfies an appropriate universal property
. The free algebra functor is left adjoint to the forgetful functor
from the category of R-algebras to the category of sets
.
Free algebras over division ring
s are free ideal ring
s.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, especially in the area of abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
known as ring theory
Ring theory
In abstract algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...
, a free algebra is the noncommutative analogue of a polynomial ring
Polynomial ring
In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...
(which may be regarded as a free commutative algebra).
Definition
For R a commutative ringCommutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
, the free (associative, unital) algebra
Algebra (ring theory)
In mathematics, specifically in ring theory, an algebra over a commutative ring is a generalization of the concept of an algebra over a field, where the base field K is replaced by a commutative ring R....
on n indeterminate
Indeterminate (variable)
In mathematics, and particularly in formal algebra, an indeterminate is a symbol that does not stand for anything else but itself. In particular it does not designate a constant, or a parameter of the problem, it is not an unknown that could be solved for, it is not a variable designating a...
s {X1,...,Xn} is the free R-module
Free module
In mathematics, a free module is a free object in a category of modules. Given a set S, a free module on S is a free module with basis S.Every vector space is free, and the free vector space on a set is a special case of a free module on a set.-Definition:...
with a basis consisting of all words over the alphabet {X1,...,Xn} (including the empty word, which is the unity of the free algebra). This R-module becomes an R-algebra
Algebra (ring theory)
In mathematics, specifically in ring theory, an algebra over a commutative ring is a generalization of the concept of an algebra over a field, where the base field K is replaced by a commutative ring R....
by defining a multiplication as follows: the product of two basis elements is the concatenation
Concatenation
In computer programming, string concatenation is the operation of joining two character strings end-to-end. For example, the strings "snow" and "ball" may be concatenated to give "snowball"...
of the corresponding words:
and the product of two arbitrary elements is thus uniquely determined (because the multiplication in an R-algebra must be R-bilinear). This R-algebra is denoted R⟨X1,...,Xn⟩. This construction can easily be generalized to an arbitrary set X of indeterminates.
In short, for an arbitrary set
, the free (associative, unital) R-algebraAlgebra (ring theory)
In mathematics, specifically in ring theory, an algebra over a commutative ring is a generalization of the concept of an algebra over a field, where the base field K is replaced by a commutative ring R....
on X is
with the R-bilinear multiplication that is concatenation on words, where X* denotes the free monoid on X (i.e. words on the letters Xi),
denotes the external direct sumDirect sum of modules
In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The result of the direct summation of modules is the "smallest general" module which contains the given modules as submodules...
, and Rw denotes the free R-module
Free module
In mathematics, a free module is a free object in a category of modules. Given a set S, a free module on S is a free module with basis S.Every vector space is free, and the free vector space on a set is a special case of a free module on a set.-Definition:...
on 1 element, the word w.
For example, in R⟨X1,X2,X3,X4⟩, for scalars α,β,γ,δ ∈R, a concrete example of a product of two elements is
.Contrast with Polynomials
Since the words over the alphabet {X1, ...,Xn} form a basis of R⟨X1,...,Xn⟩, it is clear that any element of R⟨X1, ...,Xn⟩ can be uniquely written in the form:where
are elements of R and all but finitely many of these elements are zero. This explains why the elements of R⟨X1,...,Xn⟩ are often denoted as "non-commutative polynomials" in the "variables" (or "indeterminates") X1,...,Xn; the elements are said to be "coefficients" of these polynomials, and the R-algebra R⟨X1,...,Xn⟩ is called the "non-commutative polynomial algebra over R in n indeterminates". Note that unlike in an actual polynomial ring, the variables do not commute. For example X1X2 does not equal X2X1.More generally, one can construct the free algebra R⟨E⟩ on any set E of generators
Generating set
In mathematics, the expressions generator, generate, generated by and generating set can have several closely related technical meanings:...
. Since rings may be regarded as Z-algebras, a free ring on E can be defined as the free algebra Z'E⟩.
Over a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
, the free algebra on n indeterminates can be constructed as the tensor algebra
Tensor algebra
In mathematics, the tensor algebra of a vector space V, denoted T or T•, is the algebra of tensors on V with multiplication being the tensor product...
on an n-dimensional vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
. For a more general coefficient ring, the same construction works if we take the free module
Free module
In mathematics, a free module is a free object in a category of modules. Given a set S, a free module on S is a free module with basis S.Every vector space is free, and the free vector space on a set is a special case of a free module on a set.-Definition:...
on n generators
Generating set
In mathematics, the expressions generator, generate, generated by and generating set can have several closely related technical meanings:...
.
The construction of the free algebra on E is functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
ial in nature and satisfies an appropriate universal property
Universal property
In various branches of mathematics, a useful construction is often viewed as the “most efficient solution” to a certain problem. The definition of a universal property uses the language of category theory to make this notion precise and to study it abstractly.This article gives a general treatment...
. The free algebra functor is left adjoint to the forgetful functor
Forgetful functor
In mathematics, in the area of category theory, a forgetful functor is a type of functor. The nomenclature is suggestive of such a functor's behaviour: given some object with structure as input, some or all of the object's structure or properties is 'forgotten' in the output...
from the category of R-algebras to the category of sets
Category of sets
In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets A and B are all functions from A to B...
.
Free algebras over division ring
Division ring
In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a non-trivial ring in which every non-zero element a has a multiplicative inverse, i.e., an element x with...
s are free ideal ring
Free ideal ring
In mathematics, especially in the field of ring theory, a free ideal ring, or fir, is a ring in which all left ideals are free of unique rank. A ring such that all left ideals with at most n generators is free of unique rank is called an n-fir. A semifir is a ring in which all finitely generated...
s.