## Definition ## A __commutative ring__ is a [[ring]] $(A, +, -, 0, \cdot, 1)$ with * a commutative identity for $\cdot$ $$m_\kappa:\prod_{(a:A)} \prod_{(b:A)} a\cdot b = b\cdot a$$ ## Properties ## A commutative ring is a [[commutative A3-space]] in [[abelian group]]s. ## Examples ## * Every [[contractible type]] is a commutative ring. * The [[integers]] are a commutative ring. * The [[rational numbers]] are a commutative ring. ## See also ## * [[commutative A3-space]] * [[ring]] * [[commutative cancellation ring]] * [[ideal (ring theory)]]