Division Ring

Definition

A Division Ring, also known as a skew field, is a noncommutative generalization of a field. A division ring is a ring such that every non-zero element is a unit. A notable example of a division ring is the quaternions.

Properties

Division rings share many properties with fields. However, unlike fields, division rings are not commutative. This means that the product of two elements in a division ring may depend on the order of multiplication. Despite this, division rings still possess an important property of fields: every non-zero element has a multiplicative inverse.

Examples

One of the most well-known examples of a division ring is the ring of quaternions. Quaternions extend the concept of rotation in three dimensions to four dimensions. They are used in computer graphics and control theory.

Another example of a division ring is the ring of Hamiltonian quaternions. This ring is used in quantum mechanics to describe the state of a quantum system.

A set of four-dimensional numbers represented in a three-dimensional space.

A set of four-dimensional numbers represented in a three-dimensional space.

Theorems

There are several important theorems related to division rings. One of the most significant is the Wedderburn's Little theorem, which states that every finite division ring is a field. This theorem has profound implications in the study of finite simple groups.

Another important theorem is the Artin-Wedderburn theorem, which classifies semisimple rings. This theorem is a cornerstone of noncommutative ring theory and has applications in representation theory.

Applications

Division rings have applications in various areas of mathematics and physics. In addition to their use in computer graphics and control theory, division rings are also used in algebraic topology, differential geometry, and quantum mechanics.

See Also

• Field
• Ring
• Noncommutative Ring
• Quaternion Algebra