A division ring is a nonzero ring in which every nonzero element has a multiplicative inverse. Its multiplication need not be commutative.
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A **division ring** is a type of algebraic structure in abstract algebra. It is similar to a field, but with a key difference regarding the requirement for multiplication. Here are the main characteristics of a division ring: 1. **Set with Two Operations**: A division ring consists of a set \( D \) equipped with two binary operations: addition (+) and multiplication (·).
Two ways to see it:
- a ring where inverses exist
- a field where multiplication is not necessarily commutative