To be more specific, I am looking at $F_2[D_p]$, where $D_p$ is the dihedral group of order $2p$.
If this group acts regularly on the basis of a vector space $F_2^{2p}$, and there is a subspace of $F_2^{2p}$ that is fixed under the action of the group, is it true that this subspace can be identified with a left ideal of $F_2[D_p]$? I guess the idea is that you can identify each basis element of $F_2^{2p}$ with an element in $D_p$. But is it guaranteed we can do that as long as the action is regular?
Besides, it kinda bothers me that $F_2[D_p]$ is not semi-simple, since the characteristic (which is 2) divides the order of $D_p$. This ring doesn't decompose.
The fact that the action is regular gives an equivariant isomorphism $F_2^{2p}\cong F_2[D_p]$. This isomorphism sends your invariant subspace to an invariant subspace (under left multiplication) of $F_2[D_p]$. That is, a left ideal. So, yes.