I've been studying Hopf-Galois theory, and while writing by myself certain example, I came up with this question:
Let $K$ be a field, and $\mathbb H$ the quaternion group. Is $K[\mathbb H]$ the only left ideal of the group algebra $K[\mathbb H]$?
Remember that the group algebra $K[G]$ is a $K$-space with basis the set of group elements $\{g\in G\}$, where the algebra structure is given by multiplying the basis elements using the group operation.
The reason I'm interested in this is because, to use the Generalised Fundamental Theorem of Hopf-Galois Theory, I need to determine the set of admissible coideals left ideals of $K[\mathbb H]$ (the only one of these notions I'm asking about is left ideal). But when I started thinking about it, $K[\mathbb H]$ seemed to be the only left ideal of $K[\mathbb H]$.
I don't think that result is wrong, but it feels odd (pun intended), so I decided to ask here since there may be something I'm missing or misunderstanding about the group algebra structure. Any help or hint will be appreciated, thanks in advance.
For a group $G$ with more than one element, $K[G]$ has at least three distinct left ideals: the trivial ones and the augmentation ideal.
If $K$ doesn't have characteristic $2$, the group algebra $K[H]$ (I'm using $H$ for the quaternion group with 8 elements) is semisimple and contains a nontrivial matrix ring, which has at least as many (maximal!) left ideals as elements of the field.