Division ring

A ring with identity in which every nonzero element has a multiplicative inverse. Multiplication need not be commutative.

Ring (mathematics): An algebraic structure with addition and multiplication, where addition forms an abelian group and multiplication is associative and distributes over addition. A division ring adds an identity and inverses for all nonzero elements to this broader structure.
Field (mathematics): A commutative ring with identity in which every nonzero element has a multiplicative inverse. A field satisfies the same inverse condition but also requires commutative multiplication.
Quaternion algebra: A four-dimensional algebra generated by elements whose multiplication is noncommutative. The real quaternions provide the standard finite-dimensional noncommutative division ring.
Wedderburn's little theorem: The theorem that every finite division ring is commutative. It rules out finite noncommutative division rings, making every finite example a field.
Center of a division ring: The subfield of elements that commute with every element of a division ring. The center controls scalar structure and is the base field for central division algebras.
Multiplicative identity: An element that leaves every element unchanged when multiplied on either side. The identity lets the inverse condition define a multiplicative group on nonzero elements.
Matrix ring: A ring of square matrices under matrix addition and multiplication. Matrix rings are usually noncommutative, but typically contain nonzero matrices without inverses.
Hamilton's quaternions: The real algebra generated by 1, i, j, and k, with i² = j² = k² = ijk = −1. Every nonzero quaternion has an inverse, while products such as ij and ji differ.
Skolem–Noether theorem: A theorem stating that automorphisms of a central simple algebra over its center are inner. Finite-dimensional division algebras over their centers are central simple algebras.
Central division algebra: A division algebra whose center is exactly its specified base field. These algebras organize finite-dimensional noncommutative examples over a fixed field.

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