division ring
Quick Summary
In mathematics, a ring is a set with two operations: addition and multiplication. These operations must follow specific rules, known as axioms. Addition must be commutative and associative, and there must be a zero element and additive inverses. Multiplication must be associative, and there must be a multiplicative identity (usually denoted as 1). Additionally, two distributive laws must connect addition and multiplication.1
A commutative ring is a type of ring where multiplication is also commutative, meaning the order of multiplication doesn't matter (ab = ba). The set of integers with ordinary addition and multiplication is a common example of a commutative ring.1
A division ring, also called a skew field, is a ring where multiplication is not commutative. This means that for elements a and b in the set, ab may not be equal to ba. An example of a noncommutative division ring is the set of quaternions.23
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