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.

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.

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. 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, and the addition in the ring and 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 way to connect two algebra structures. It keeps the rules for adding, multiplying, and scaling by numbers the same. When we have two algebra structures, a homomorphism makes sure everything fits together properly.

These homomorphisms help us see how different algebra structures are related, forming a bigger system called a category. This category includes all the 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.

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. It still follows all the same rules, like adding, multiplying, and scaling up.

We can also create new algebras by taking pieces away. This is called a quotient algebra. We can also combine many algebras together in different ways. Some of these ways are called direct products, free products, and tensor products.

The free algebra is made from symbols. 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 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.

For example, if we take the ring of continuous functions on a group, this ring is an associative algebra. The dual of this algebra can also be 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.

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 type of algebra where its parts fit together in a neat and tidy way. This makes it easier to study and learn about the algebra's features. When an algebra is separable, it works well with certain operations, which helps mathematicians use it more easily.

Finite-dimensional algebra

See also: Central simple algebra

A finite-dimensional algebra over a field has special properties. If the algebra is commutative, it can be broken down into simpler parts. These parts are either fields or have nice structures.

For noncommutative algebras, things are more complex. A simple algebra looks like a set of matrices over a division algebra. When we combine these, we get the Artin–Wedderburn theorem. This theorem helps us understand complex algebras by breaking them into smaller, easier 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, there are smaller, neatly arranged sets called lattices. These lattices follow certain rules.

We can also talk about orders. Orders 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 can be an order. A maximal order is the largest order possible in a given space.

Related concepts

Coalgebras

Main article: Coalgebra

An associative algebra is a special kind of mathematical space. It has rules for adding and multiplying its elements. These rules can be described using ideas from category theory. Category theory helps organize mathematical structures. By looking at these rules in a different way—by reversing certain maps—we can understand 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 a way to show how the algebra works inside another 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 seeing how they work together. But this can be tricky. Mathematicians sometimes use special structures like Hopf algebras or Lie algebras to help make everything work together better.

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

Non-unital algebras

Some writers talk about "associative algebras" without needing a special number called an identity for multiplication. They study 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.