The \(R\)-points of \(\mathbb{G}_a\) over \(\operatorname{Spec} k\) are morphisms \(\operatorname{Spec} R \to \mathbb{G}_{a, \operatorname{Spec} k}\) over \(\operatorname{Spec} k\). By the affine-scheme equivalence and the construction \(\mathbb{G}_{a, \operatorname{Spec} k} = \operatorname{Spec} k[T]\), such morphisms are in bijection with \(k\)-algebra homomorphisms \(k[T] \to R\). By the universal property of the polynomial ring, these are in bijection with elements of \(R\) (the image of \(T\)):

\[ \mathbb{G}_a(R) \;\cong\; \operatorname{Hom}_{k\text{-Alg}}(k[T], R) \;\cong\; R. \]

To verify that the bijection is a group isomorphism, evaluate the multiplication \(\mathbb{G}_a(R) \times \mathbb{G}_a(R) \to \mathbb{G}_a(R)\) on pairs \((r, s) \in R \times R\). This multiplication is induced by the comultiplication \(\Delta(T) = T \otimes 1 + 1 \otimes T\) of the coordinate Hopf algebra: the image of \(T\) under the composed algebra hom \(k[T] \to k[T] \otimes_k k[T] \to R\) (first map \(\Delta\), second map sending the first factor to \(r\) and the second to \(s\)) is \(r + s\). The identity element corresponds to \(\varepsilon(T) = 0 \in R\), and inversion to \(S(T) = -T \in R\). All three coincide with the additive group structure on \(R\).
\(\square\)