Free algebra

Short description: Free object in the category of associative algebras

This article is about free algebras in ring theory. For the more general free algebras in universal algebra, see Free object.

Algebraic structure → Ring theory
Ring theory

Basic concepts

Rings

• Subrings
• Ideal
• Quotient ring

• Fractional ideal
• Total quotient ring
• Product ring
• Free product of associative algebras
• Tensor product of rings

Ring homomorphisms

• Kernel
• Inner automorphism
• Frobenius endomorphism

Algebraic structures

• Module
• Associative algebra
• Graded ring
• Involutive ring
• Category of rings

• Initial ring ℤ
• Terminal ring 0=ℤ1

Related structures

• Field

• Finite field
• Non-associative ring

• Lie ring
• Jordan ring
• Semiring

• Semifield

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Commutative algebra

Commutative rings

• Integral domain

• Integrally closed domain
• GCD domain
• Unique factorization domain
• Principal ideal domain
• Euclidean domain
• Field

• Finite field
• Composition ring

• Polynomial ring
• Formal power series ring

Algebraic number theory

• Algebraic number field
• Ring of integers
• Algebraic independence
• Transcendental number theory
• Transcendence degree

p-adic number theory and decimals

• Direct limit/Inverse limit
• Zero ring ℤ1
• Integers modulo pn ℤ/pnℤ
• Prüfer p-ring ℤ(p∞)
• Base-p circle ring 𝕋
• Base-p integers ℤ
• p-adic rationals ℤ[1/p]
• Base-p real numbers ℝ
• p-adic integers ℤp
• p-adic numbers ℚp
• p-adic solenoid 𝕋p

Algebraic geometry

• Affine variety

* * *

Noncommutative algebra

Noncommutative rings

• Division ring
• Semiprimitive ring
• Simple ring
• Commutator

Noncommutative algebraic geometry

Free algebra

Clifford algebra

• Geometric algebra
Operator algebra

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In mathematics, especially in the area of abstract algebra known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring since its elements may be described as "polynomials" with non-commuting variables. Likewise, the polynomial ring may be regarded as a free commutative algebra.

## Contents

* 1 Definition
* 2 Contrast with polynomials
* 3 See also
* 4 References

## Definition

For R a commutative ring, the free (associative, unital) algebra on n indeterminates {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 unit 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:

(Xi1Xi2⋯Xil)⋅(Xj1Xj2⋯Xjm)=Xi1Xi2⋯XilXj1Xj2⋯Xjm,

and the product of two arbitrary R-module 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 X={Xi;i∈I}, the free (associative, unital) R-algebra on X is

R⟨X⟩:=⨁w∈X∗Rw

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

(αX1X22+βX2X3)⋅(γX2X1+δX14X4)=αγX1X23X1+αδX1X22X14X4+βγX2X3X2X1+βδX2X3X14X4.

The non-commutative polynomial ring may be identified with the monoid ring over R of the free monoid of all finite words in the Xi.

## 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 written uniquely in the form:

∑k=0∞∑i1,i2,⋯,ik∈{1,2,⋯,n}ai1,i2,⋯,ikXi1Xi2⋯Xik,

where ai1,i2,...,ik 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 ai1,i2,...,ik 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 functorial 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 rings are free ideal rings.

## See also

* Cofree coalgebra
* Tensor algebra
* Free object
* Noncommutative ring
* Rational series
* Term algebra

## References

* Berstel, Jean; Reutenauer, Christophe (2011). Noncommutative rational series with applications. Encyclopedia of Mathematics and Its Applications. 137. Cambridge: Cambridge University Press. ISBN 978-0-521-19022-0.
* Hazewinkel, Michiel, ed. (2001), "Free associative algebra", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=f/f041520

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