Positive characteristic analogues of finite algebraic numbers

J.~Rosen introduced the ring ${\mathcal{P}}_{\mathcal{A}}^0$ of so-called finite algebraic numbers, which may be seen as an analogue of certain periods in the ring $\mathcal{A}={\prod }_p\mathbb{Z}\mathrm{/}p\mathbb{Z}\mathrm{/}{⨁}_p\mathbb{Z}\mathrm{/}p\mathbb{Z}$, $p$ running through all prime numbers. In this article, we introduce its positive characteristic analogue ${\mathcal{P}}_{{\mathcal{A}}_K}^0$ over the rational function field $K={\mathbb{F}}_q(\theta )$, $q$ being a prime power, and study foundational properties.