## Definition ## A __ring__ is an [[abelian group]] $R$ with an element $1:R$ and a [[abelian group homomorphism]] $\alpha:R \to (R \times R)$ such that * $\alpha(1) = \mathrm{id}_R$ * for all $a:R$, $\alpha(a)(1) = a$ * for all $a:R$, $b:R$, and $c:R$, $(\alpha(a) \circ \alpha(b))(c) = \alpha(a)(\alpha(b)(c))$ We define the [[bilinear function]] $(-)\cdot(-):M \times M \to M$ as $$a \cdot b \coloneqq \alpha(a)(b)$$ ## Examples ## * Every [[contractible type]] is a ring. * The [[integers]] are a ring. * The [[rational numbers]] are a ring. ## See also ## * [[abelian group]] * [[monoid]] * [[commutative ring]]