Let $X$ be a smooth connected commutative algebraic group over an algebraic closed field $K$. Is it true that its ring of endomorphisms (in the category of algebraic groups), denoted End($X$), always (left or right) Noetherian?
It is well known that if $X$ is an abelian variety then End($X$) is a free abelian group of finite rank. It follows that End($X$) is Noetherian as a ring.
If $X$ is a connected commutative complex linear algebraic group then $X\simeq \mathbb{G}_a^r \times \mathbb{G}_m^s$ for some $r,s \in \mathbb{N}$. Using this, we can see that End$(X)$ is also a Noetherian ring.
What about the general case? Can we use Chevalley's structure theorem for algebraic groups to say something about End($X$)?
Thank you in advance.