Simple and semisimple rational representations

Let $k$ be a field, let $G$ be an affine group scheme of finite type over $k$, and let $(V,r)$ be a rational representation of $G$, with subrepresentations the subcomodules of $V$ (Rational representations and comodules of an affine group scheme). The representation $(V,r)$ is simple (or irreducible) if $V\mathrm{\ne }0$ and the only subrepresentations of $V$ are $0$ and $V$. It is semisimple (or completely reducible) if $V$ is an internal direct sum of simple subrepresentations, $V={⨁}_{i\in I}{S}_i,S_i\subseteq V\text{ simple};$ the zero representation is semisimple, being the empty direct sum.

• Terminology. Simple and semisimple representations are traditionally called irreducible and completely reducible when regarded as representations; the two pairs of words are synonyms here, as in the source.
• Finite dimensionality. Every simple rational representation of an affine finite-type group scheme is finite-dimensional; this is proved on the same page and is not part of the definition.
• Subrepresentations. Under the correspondence of the cited definition, subrepresentations are exactly the subcomodules, so simplicity and semisimplicity can be checked on comodules; no smoothness of $G$ is required, and no choice principle is used in the definition.