Let $R[x_1, x_2, \dots]$ be a ring of formal polynomials in a countably $\infty$ number of indeterminates $x_i$, over a commutative ring $R$.
The commutative monoid $X = \{ x^e = x_1^{e_1} x_2^{e_2} \cdots \vert e_i \in \Bbb{N}, \ e_i \neq 0 \text{ only finitely many times }\}$ is isomorphic to $(\Bbb{N}_{\gt 0}, \cdot)$. An ismorphism is given by any bijection between $\{x_1, x_2, \dots\}$ and the primes $\{p_1, p_2, \dots \} \subset \Bbb{N}_{\gt 0}$ extended homomorphically. Stated another way, any commutative free monoid on a countably $\infty$ set is isomorphic to $\Bbb{N}_{\gt 0}$.
So let $f: X \to \Bbb{N}_{\gt 0}$ be any such isomorphism. Then the order on $X$ defined as $a \lt b \iff f(a) \lt f(b)$ is an admissible ordering (one that can be used for Gröbner bases).
Additionally, since $X$ is countable we have that $R[x_1, \dots]$ is an $R$-module with countable basis $X$.