| context | $\langle X,+,* \rangle \in \mathrm{ring}(X)$ |
| definiendum | $\langle X,+,* \rangle \in \mathrm{it}$ |
| postulate | $\langle X,* \rangle \in \mathrm{monoid}(X)$ |
The second requirement implies that there is an identiy for the binary operation $*$.
One generally (also) calls $X$ the unital ring, i.e. the set where the operations “$+$” and “$*$” are defined on.
Wikipedia: Ring