Law Packages for the Operations of ℕ₀ and of a Commutative Ring

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, with $\mathbb{N}_{0}$, $0$ and $1$ as in The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion (ii):

(i) The natural numbers with zero. The addition $+$ of $\mathbb{N}_{0}$ is an associative and commutative binary operation on $\mathbb{N}_{0}$ with neutral element $0$, and the multiplication $\cdot$ of $\mathbb{N}_{0}$ is an associative and commutative binary operation on $\mathbb{N}_{0}$ with neutral element $1$.

(ii) Commutative rings. If $R$, with $+$, $\cdot$, $0$ and $1$, is a commutative ring, then $+$ is an associative and commutative binary operation on $R$ with neutral element $0$, and $\cdot$ is an associative and commutative binary operation on $R$ with neutral element $1$.