Algebra and Its Meanings

The Japanese translation of “algebra” is “代数,” which typically evokes the concept of “algebra (代数学)” in the context of mathematics, such as basic algebraic equations and operations. However, the term “algebra” has a broader and more specialized meaning within higher mathematics and abstract algebra.

In this other sense, “algebra” is translated in Japanese as either “代数” or “環.” The 環 here is not an ordinary ring, but an algebra. In this expanded sense, the term “algebra” is sometimes translated into Japanese as either “代数” or “環.” Here, “環” does not refer to the general concept of rings as they are typically understood in abstract algebra but instead to a particular kind of algebraic structure. Algebras is also called associative algebras.

Definition

A vector space \( A \) over a field \( K \) is called an algebra over \( K \) if \( A \) has a ring structure and the mapping that defines the product \( A \times A \to A \) is bilinear.

Examples

A simple example of an algebra is the matrix algebra, which is one of the operator algebras. In operator algebra theory, the most important algebras are \( C^{\ast} \)-algebras and von Neumann algebras. Algebras that do not satisfy the associative law are called non-associative algebras, with Lie algebras and Jordan algebras being prominent examples.