# Associative algebra

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In mathematics, an **associative algebra** *A* is an associative ring that has a compatible structure of a vector space over a certain field *K* or, more generally, of a module over a commutative ring *R*. Thus *A* is endowed with binary operations of addition and multiplication satisfying a number of axioms, including associativity of multiplication and distributivity, as well as compatible multiplication by the elements of the field *K* or the ring *R*.

In some areas of mathematics, associative algebras are typically assumed to have a multiplicative unit, denoted 1. To make this extra assumption clear, these associative algebras are called unital algebras.

## Formal definition

Let *R* be a fixed commutative ring. An **associative R-algebra** is an additive abelian group

*A*which has the structure of both a ring and an

*R*-module in such a way that ring multiplication is

*R*-bilinear:

for all *r* ∈ *R* and *x*, *y* ∈ *A*.
We say *A* is **unital** if it contains an element 1 such that

for all *x* ∈ *A*.

If *A* itself is commutative (as a ring) then it is called a **commutative R-algebra**.

### From *R*-modules

Starting with an *R*-module *A*, we get an associative *R*-algebra by equipping *A* with an *R*-bilinear mapping *A* × *A* → *A* such that

for all *x*, *y*, and *z* in *A*. This *R*-bilinear mapping then gives *A* the structure of a ring and an associative *R*-algebra. Every associative *R*-algebra arises this way.

Moreover, the algebra *A* built this way will be unital if and only if

This definition is equivalent to the statement that a unital associative *R*-algebra is a monoid in ** R-Mod** (the monoidal category of

*R*-modules).

### From rings

Starting with a ring *A*, we get a unital associative *R*-algebra by providing a ring homomorphism whose image lies in the center of *A*. The algebra *A* can then be thought of as an *R*-module by defining

for all *r* ∈ *R* and *x* ∈ *A*.

If *A* is commutative then the center of *A* is equal to *A*, so that a commutative *R*-algebra can be defined simply as a homomorphism of commutative rings.

## Algebra homomorphisms

A homomorphism between two associative *R*-algebras is an *R*-linear ring homomorphism. Explicitly, is an **associative algebra homomorphism** if

For a homomorphism of *unital* associative *R*-algebras, we also demand that

The class of all unital associative *R*-algebras together with algebra homomorphisms between them form a category, sometimes denoted ** R-Alg**.

The subcategory of commutative *R*-algebras can be characterized as the coslice category *R*/**CRing** where **CRing** is the category of commutative rings.

## Examples

- The square
*n*-by-*n*matrices with entries from the field*K*form a unitary associative algebra over*K*. - The complex numbers form a 2-dimensional unitary associative algebra over the real numbers.
- The quaternions form a 4-dimensional unitary associative algebra over the reals (but not an algebra over the complex numbers, since if complex numbers are treated as a subset of the quaternions, complex numbers and quaternions do not commute).
- The 2 × 2 real matrices form an associative algebra useful in plane mapping.
- The polynomials with real coefficients form a unitary associative algebra over the reals.
- Given any Banach space
*X*, the continuous linear operators*A*:*X*→*X*form a unitary associative algebra (using composition of operators as multiplication); this is a Banach algebra. - Given any topological space
*X*, the continuous real- or complex-valued functions on*X*form a real or complex unitary associative algebra; here the functions are added and multiplied pointwise. - An example of a non-unitary associative algebra is given by the set of all functions
*f*:**R**→**R**whose limit as*x*nears infinity is zero. - The Clifford algebras, which are useful in geometry and physics.
- Incidence algebras of locally finite partially ordered sets are unitary associative algebras considered in combinatorics.
- Any ring
*A*can be considered as a**Z**-algebra in a unique way. The unique ring homomorphism from**Z**to*A*is determined by the fact that it must send 1 to the identity in*A*. Therefore rings and**Z**-algebras are equivalent concepts, in the same way that abelian groups and**Z**-modules are equivalent. - Any ring of characteristic
*n*is a (**Z**/*n***Z**)-algebra in the same way. - Any ring
*A*is an algebra over its center*Z*(*A*), or over any subring of its center. - Any commutative ring
*R*is an algebra over itself, or any subring of*R*. - Given an
*R*-module*M*, the endomorphism ring of*M*, denoted End_{R}(*M*) is an*R*-algebra by defining (*r*·φ)(*x*) =*r*·φ(*x*). - Any ring of matrices with coefficients in a commutative ring
*R*forms an*R*-algebra under matrix addition and multiplication. This coincides with the previous example when*M*is a finitely-generated, free*R*-module. - Every polynomial ring
*R*[*x*_{1}, ...,*x*_{n}] is a commutative*R*-algebra. In fact, this is the free commutative*R*-algebra on the set {*x*_{1}, ...,*x*_{n}}. - The free
*R*-algebra on a set*E*is an algebra of polynomials with coefficients in*R*and noncommuting indeterminates taken from the set*E*. - The tensor algebra of an
*R*-module is naturally an*R*-algebra. The same is true for quotients such as the exterior and symmetric algebras. Categorically speaking, the functor which maps an*R*-module to its tensor algebra is left adjoint to the functor which sends an*R*-algebra to its underlying*R*-module (forgetting the ring structure). - Given a commutative ring
*R*and any ring*A*the tensor product*R*⊗_{Z}*A*can be given the structure of an*R*-algebra by defining*r*·(*s*⊗*a*) = (*rs*⊗*a*). The functor which sends*A*to*R*⊗_{Z}*A*is left adjoint to the functor which sends an*R*-algebra to its underlying ring (forgetting the module structure).

## Constructions

- Subalgebras
- A subalgebra of an
*R*-algebra*A*is a subset of*A*which is both a subring and a submodule of*A*. That is, it must be closed under addition, ring multiplication, scalar multiplication, and it must contain the identity element of*A*. - Quotient algebras
- Let
*A*be an*R*-algebra. Any ring-theoretic ideal*I*in*A*is automatically an*R*-module since*r*·*x*= (*r*1_{A})*x*. This gives the quotient ring*A*/*I*the structure of an*R*-module and, in fact, an*R*-algebra. It follows that any ring homomorphic image of*A*is also an*R*-algebra. - Direct products
- The direct product of a family of
*R*-algebras is the ring-theoretic direct product. This becomes an*R*-algebra with the obvious scalar multiplication. - Free products
- One can form a free product of
*R*-algebras in a manner similar to the free product of groups. The free product is the coproduct in the category of*R*-algebras. - Tensor products
- The tensor product of two
*R*-algebras is also an*R*-algebra in a natural way. See tensor product of algebras for more details.

## Associativity and the multiplication mapping

Associativity was defined above quantifying over all *elements* of *A*. It is possible to define associativity in a way that does not explicitly refer to elements. An algebra is defined as a map *M* (multiplication) on a vector space *A*:

An associative algebra is an algebra where the map *M* has the property

Here, the symbol refers to function composition, and Id : *A* → *A* is the identity map on *A*.

To see the equivalence of the definitions, we need only understand that each side of the above equation is a function that takes three arguments. For example, the left-hand side acts as

Similarly, a unital associative algebra can be defined in terms of a unit map

which has the property

Here, the unit map η takes an element *k* in *K* to the element *k1* in *A*, where *1* is the unit element of *A*. The map *s* is just plain-old scalar multiplication: ; thus, the above identity is sometimes written with Id standing in the place of *s*, with scalar multiplication being implicitly understood.

## Coalgebras

An associative unitary algebra over *K* is based on a morphism *A*×*A*→*A* having 2 inputs (multiplicator and multiplicand) and one output (product), as well as a morphism *K*→*A* identifying the scalar multiples of the multiplicative identity. These two morphisms can be dualized using categorial duality by reversing all arrows in the commutative diagrams which describe the algebra axioms; this defines the structure of a coalgebra.

There is also an abstract notion of F-coalgebra.

## Representations

A representation of an algebra is a linear map ρ: *A* → gl(*V*) from *A* to the general linear algebra of some vector space (or module) *V* that preserves the multiplicative operation: that is, ρ(*xy*)=ρ(*x*)ρ(*y*).

Note, however, that there is no natural way of defining a tensor product of representations of associative algebras, without somehow imposing additional conditions. Here, by *tensor product of representations*, the usual meaning is intended: the result should be a linear representation on the product vector space. Imposing such additional structure typically leads to the idea of a Hopf algebra or a Lie algebra, as demonstrated below.

### Motivation for a Hopf algebra

Consider, for example, two representations and . One might try to form a tensor product representation according to how it acts on the product vector space, so that

However, such a map would not be linear, since one would have

for *k* ∈ *K*. One can rescue this attempt and restore linearity by imposing additional structure, by defining a map Δ: *A* → *A* × *A*, and defining the tensor product representation as

Here, Δ is a comultiplication. The resulting structure is called a bialgebra. To be consistent with the definitions of the associative algebra, the coalgebra must be co-associative, and, if the algebra is unital, then the co-algebra must be unital as well. Note that bialgebras leave multiplication and co-multiplication unrelated; thus it is common to relate the two (by defining an antipode), thus creating a Hopf algebra.

### Motivation for a Lie algebra

One can try to be more clever in defining a tensor product. Consider, for example,

so that the action on the tensor product space is given by

This map is clearly linear in *x*, and so it does not have the problem of the earlier definition. However, it fails to preserve multiplication:

But, in general, this does not equal

Equality would hold if the product *xy* were antisymmetric (if the product were the Lie bracket, that is, ), thus turning the associative algebra into a Lie algebra.

## References

- {{#invoke:citation/CS1|citation

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- Ross Street,
*Quantum Groups: an entrée to modern algebra*(1998).*(Provides a good overview of index-free notation)*

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