It occurred to me that since the group algebra of $\mathbb{Z}$, $k\mathbb{Z}$, has multiplication $$\left(\sum _{n\in \mathbb Z} a_n n\right)\cdotp \left(\sum _{n\in \mathbb Z} b_n n\right)=\left(\sum _{n\in \mathbb Z} \sum _{k\in \mathbb{Z}} a_k b_{n-k} n\right)$$ we can think of the polynomial algebra $k\left[X\right]$ as a subalgebra of $k\mathbb{Z}$ spanned by the non-negative elements.
Since the group algebra construction works for monoids just as well, can we then think of the polynomial algebra $k\left[X\right]$ as $k\mathbb{N}$ as the "monoid algebra" of the natural numbers? This would be quite a neat definition of the polynomial algebra, if it works, I think.