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
Every contractible type is a ring.
The integers are a ring.
The rational numbers are a ring.