Associative algebra

In mathematics, an associative algebra is a special kind of mathematical structure. It is built over a commutative ring, which is often a field. This structure has addition, multiplication, and a special kind of multiplication called scalar multiplication. Together, these operations follow certain rules that make the structure both a ring and a kind of space called a module or vector space.

One common example of an associative algebra is the ring of square matrices over a commutative ring. These matrices are multiplied in the usual way, following specific rules. When the multiplication in an associative algebra is commutative—meaning the order of multiplication does not change the result—it is called a commutative algebra.

Associative algebras usually have a special element called a multiplicative identity, often written as 1. This identity element behaves in a way similar to the number 1 in regular arithmetic, ensuring that multiplying by it does not change the value. Not all structures have this identity, and those without are called non-unital associative algebras. Every ring can also be seen as an associative algebra in a particular way.

Definition

Let R be a commutative ring (which could also be a field). An associative R-algebra A (or simply an R-algebra A) is a special kind of ring. It is also an R-module in a way that the addition in the ring and the addition in the module are the same. There is also a way to multiply elements of R by elements of A, and this multiplication follows certain rules.

Another way to think about it is that an associative algebra A is a ring that has a special mapping from R into the middle of A. This mapping helps us understand how elements of R interact with elements of A.

Every ring can be seen as an associative Z-algebra, where Z stands for the ring of the integers.

A commutative algebra is an associative algebra where the multiplication of elements also follows a commutative rule, meaning the order of multiplication does not change the result.

Algebra homomorphisms

Main article: algebra homomorphism

An algebra homomorphism is a special kind of map between two algebra structures. It keeps the important rules of addition, multiplication, and scaling by numbers the same. When we have two algebra structures, a homomorphism between them follows certain rules to make sure everything lines up correctly.

These homomorphisms help us understand how different algebra structures relate to each other, forming a bigger system called a category. This category includes all these algebra structures and the special maps between them.

Examples

The simplest example of an associative algebra is a ring by itself. Any ring can be seen as an algebra over its center or any smaller ring inside the center. This means that rings and certain types of algebras are very closely related.

Other examples include:

• Any ring of square matrices with numbers in a field forms an algebra.
• The complex numbers are a type of algebra over the real numbers.
• Polynomial rings are algebras where you can add and multiply polynomials.
• In analysis, the continuous functions on a space or the operators between spaces can form algebras.
• In geometry and physics, structures like Clifford algebras and Poisson algebras are important examples.

Constructions

A subalgebra is a smaller part of a bigger algebra that still follows all the same rules, like adding, multiplying, and scaling up.

We can also create new algebras by taking pieces away, called quotient algebras, or by combining many algebras together in different ways, like direct products, free products, and tensor products.

The free algebra is one made from symbols, and if we make sure everything commutes, we get a special kind of algebra called a polynomial algebra.

Dual of an associative algebra

When we have a special kind of math structure called an associative algebra over a ring, we can look at something called its dual module. Sometimes, this dual module can also be an associative algebra if the original algebra has extra features.

For example, if we take the ring of continuous functions on a certain group, this ring is an associative algebra and also has special actions called co-multiplication and co-unit. Because of these, the dual of this algebra is also an associative algebra. These special actions help when we want to combine representations of associative algebras.

Main article: § Representations

Enveloping algebra

See also: Non-associative algebra § Associated algebras

In math, if we have a special kind of structure called an associative algebra over a ring, we can create something called its enveloping algebra. This is made by combining the algebra with a version of itself that has its operations reversed, depending on how the writer prefers to do it.

A special kind of structure called a bimodule over this algebra is the same as a left module over its enveloping algebra.

Separable algebra

Main article: Separable algebra

A separable algebra is a special kind of algebra where the way its parts fit together is very neat and organized. This makes it easier to study and understand the algebra's properties. When an algebra is separable, it behaves nicely with certain operations, which helps mathematicians work with it more simply.

Finite-dimensional algebra

See also: Central simple algebra

When we talk about a finite-dimensional algebra over a field, it has special properties. If the algebra is commutative, it breaks down into simpler pieces that are easy to study. These pieces are either fields or have other nice structures.

For noncommutative algebras, things get more interesting. A simple algebra looks like a set of matrices over another type of algebra called a division algebra. When we put these together, we get what’s known as the Artin–Wedderburn theorem, which helps us understand more complex algebras by breaking them into smaller, easier-to-handle parts.

Lattices and orders

Main article: Order (ring theory)

Imagine you have a special kind of math space called a vector space. Inside this space, we can find smaller, neatly arranged sets called lattices. These lattices follow certain rules and fit perfectly within the larger space.

We can also talk about orders, which are special types of lattices that follow extra rules. Sometimes there are fewer orders than lattices. For example, in simple number systems, not every lattice meets the requirements to be an order. A maximal order is the biggest order possible within a given space.

Related concepts

Coalgebras

Main article: Coalgebra

An associative algebra is a special kind of mathematical space that has rules for adding and multiplying its elements. These rules can be described in a different way using ideas from category theory, which helps organize mathematical structures. By looking at these rules in a "dual" way—by reversing the directions of certain maps—we can also understand the structure of something called a coalgebra. There is also a more general idea called an F-coalgebra, which is loosely connected to the idea of a coalgebra.

Representations

Main article: Algebra representation

A representation of an algebra is like a way to show how the algebra works inside another mathematical space called a vector space. Think of it as a map that connects the algebra to actions you can do on vectors, like stretching or flipping them.

When you have two different algebras and want to combine their representations, you can sometimes do this by looking at how they work together. But this isn’t always simple, and it can get tricky if you just try to combine them directly. To fix this, mathematicians sometimes use special structures like Hopf algebras or Lie algebras, which add extra rules to make everything work nicely together.

These structures help organize how different parts of the algebra interact, making sure the combined actions still follow the algebra’s original rules.

Non-unital algebras

Some writers talk about "associative algebras" in a way that does not always need a special number called an identity for multiplication. They look at maps that may not always keep this special number unchanged.

An example of a non-unital associative algebra is the group of all functions from the real numbers to the real numbers that get very small as the numbers get really big.

Another example is the space of smooth repeating functions, using a special kind of multiplication called convolution.