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For $G={\mathbb{G}}_a$, write $k[G]=k[t]$. Under the left-right regular representation of an affine algebraic group,so the action factors through the difference map ${\mathbb{G}}_a\times {\mathbb{G}}_a\to {\mathbb{G}}_a$. The degree filtrationis stable and has trivial one-dimensional successive quotients. The module is indecomposable; every nonzero submodule contains a nonzero translation difference of lower degree and, on iteration using suitable translations, meets the unique invariant line $k\cdot 1$.

For every finite-dimensional rational $G$-module $V$, the matrix-coefficient construction givesOn the other hand,Indeed, a nonzero finite-dimensional quotient of $k[G]$ would have a simple quotient. Every simple rational representation of the unipotent algebraic group ${\mathbb{G}}_a$ is trivial, so this would give a nonzero translation-invariant functional on $k[t]$. No such functional exists: if $c_j$ is its first nonzero value on a monomial, translating a sufficiently high monomial produces a nonzero coefficient times $c_j$, contradicting invariance.