It is well-known that the group of units in the $2$-adic integers $\mathbb{Z}_2$ is isomorphic to the product of a finite abelian group with $\mathbb{Z}_2$. This fact generalizes to the ring of integers $\mathcal{O}$ of a finite separable extension $K$ of $\mathbb{Q}_2$: the group of units in $\mathcal{O}$ is isomorphic to the product of a finite abelian group with $\mathbb{Z}_2^{[K : \mathbb{Q}_2]}$.
Question: What is the structure of the units in the ring $R = \mathbb{Z}_2[x]/(x^2 - 4)$? Notice that this ring is not the ring of integers in its algebra of fractions: the fraction algebra of $R$ is $\mathbb{Q}_2 \times \mathbb{Q}_2$, so the ring of integers is $\mathbb{Z}_2 \times \mathbb{Z}_2$, but $R$ is evidently not equal to $\mathbb{Z}_2 \times \mathbb{Z}_2$.
What I Know: Every unit $u \in R^\times$ can be expressed as $u = r \cdot (1 + d\cdot x)$, where $r$ is a unit in $\mathbb{Z}_2$ and $d \in \mathbb{Z}_2$. The identification $R^\times \to \mathbb{Z}_2^\times \times \mathbb{Z}_2$ defined by $u \mapsto (r, d)$ is a bijection of sets. But this identification is not a group homomorphism, and I'm not sure how to determine the structure of $R^\times$ as a group.
