Commutative ring

A ring $R$ (Ring: an abelian group under addition and a monoid under multiplication, with multiplication distributing over addition on both sides) is commutative when its multiplication is commutative (Binary operation on a set; associativity, commutativity, and a subset closed under the operation):

$xy \;=\; yx \qquad \text{for all } x, y \in R .$

Addition is commutative in every ring, by axiom (R1), so the word refers to multiplication alone.

• One of the two distributive laws becomes redundant. In a commutative ring the right law $(y+z)x = yx + zx$ follows from the left one, since $(y+z)x = x(y+z) = xy + xz = yx + zx$. Both are still stated in Ring: an abelian group under addition and a monoid under multiplication, with multiplication distributing over addition on both sides, because a ring is not assumed commutative there.

• Every field is a commutative ring (Every field is a commutative ring with $1 \ne 0$; it is an integral domain, and it is a commutative division ring), and $\mathbb{Z}$, $\mathbb{Q}$ and $\mathbb{R}$ are commutative rings; the companion page records each of those as an instance. The quaternions are a ring that is not commutative (The quaternions $\mathbb{H}$: real quadruples with componentwise addition and an explicit multiplication formula matching the table on $1, i, j, k$, $\mathbb{H}$ is a division ring that is not commutative, hence not a field: $q^{-1} = \bar q / N(q)$ for $q \ne 0$, while $ij = k$ and $ji = -k$).