I am interested in the following situation.
Let $p, q$ be two distinct primes.
Let $G$ be an elementary Abelian p-group, say $G = G_1 \oplus \cdots \oplus G_\ell$ for cyclic groups $G_i$ of order $p$, $\ell \geq 1$.
Now consider the group ring $\mathbb F_q[G]$. I am interested in the structure of this ring. What I already found about this is the following:
(1) $\mathbb F_q[G]$ is a semisimple, commutative ring due to Maschke's Theorem and it can be written as a direct sum of finite fields. Moreover, these fields have to be of characteristic q.
(2) Since $G$ is finite and Abelian, $\mathbb F_q[G]$ is a finite commutative ring, and can thus be decomposed into local rings $$\mathbb F_q[G] = e_0\mathbb F_q[G] \oplus \cdots \oplus e_k\mathbb F_q[G],$$ for certain idempotent, non-trivial, pairwise orthogonal elements $e_i$. First question: I guess (1) implies that these local rings are finite fields?
My main question is the following: Is there any way in which the size of the finite fields in the decomposition of $\mathbb F_q[G]$ can be bounded, in terms of $p,q$ only and independent of $\ell$? How do these finite fields look like? They are of the form $\mathbb F_q[X]/f(X)$, but where does the polynomial $f$ come from in this particular case? Or in other words, is there a simple way to express the elements $e_i$ in the decomposition into local rings? In which way does the decomposition of $G$ into cyclic $p$-groups occur in the decomposition of $\mathbb F_q[G]$?
I am very grateful for hints and ideas about any of these points. Thanks a lot!