What can be said about "splitting rings" for finite groups? Let $G$ be a finite group. Classically, a splitting field of $G$ is a field $K$ such that the group algebra $K[G]$ is isomorphic to a direct product of matrix algebras $M_n(K)$. I want to know what can be said about splitting rings of $G$; that is, commutative rings $R$ such that the group algebra $R[G]$ is isomorphic to a direct product of matrix algebras $M_n(R)$. These correspond to rings over which the representation theory of $G$ (say, acting on finite free $R$-modules) "looks the same as over an algebraically closed field" (of characteristic not dividing $|G|$).
What is clear from general principles is the following: first, $\overline{\mathbb{Q}}$ itself is a splitting field, and second, the isomorphism $\overline{\mathbb{Q}}[G] \cong \prod_i M_{n_i}(\overline{\mathbb{Q}})$ is defined over a finitely generated subring of $\overline{\mathbb{Q}}$, which is contained in the ring of integers of some number field localized at finitely many primes. So we can take $R$ to be what I believe is called a ring of $S$-integers.

Question 1: Can $R$ be taken to be the ring of integers of some number field localized only at primes dividing $|G|$?

The motivation is that this would give a satisfying (to my mind) answer to this question about relating the representation of $G$ over $\mathbb{C}$ and over $\overline{\mathbb{F}_p}$ where $p$ is a prime not dividing $|G|$, since $R$ would then admit homomorphisms to all of these fields. As mentioned in that question, the representation theory of $G$ is "the same" over all these fields, and finding representations over a ring $R$ admitting homomorphisms to all of them would be a good way to make this completely precise.

Question 2: Can $R$ can be taken to be $\mathbb{Z}[\zeta_{|G|}][|G|^{-1}]$?

I can motivate this choice in at least two ways: first, this choice works and I think is best possible for $G = C_n$ the finite cyclic groups. Second, over this ring we can define the idempotents $e_V = \frac{\chi_V(1)}{|G|} \sum_{g \in G} \chi_V(g) g^{-1}$ which, acting on a representation of $G$, project it to its isotypic components, and which, as elements of the group algebra, are central and so decompose it as a direct product of some mysterious rings. These rings, after extension of scalars to a splitting field, become matrix algebras, but I don't understand them integrally; in particular I don't understand what modules over them look like, which I would like to to get the best possible comparison between the representation of $G$ in characteristic zero and in positive characteristic. Here is my most optimistic guess about how these rings behave:

Question 3a: Are the direct factors corresponding to the idempotents $e_V$ above (equivalently, the quotients of the group algebra $\mathbb{Z}[\zeta_{|G|}][|G|^{-1}]$ by the ideals $(1 - e_V)$) Azumaya algebras over $\mathbb{Z}[\zeta_{|G|}][|G|^{-1}]$?


Question 3b: Regardless of the answer to Question 3a, is every finitely generated projective module over these direct factors a direct sum of copies of a single module which, after base change to a splitting field $K$, becomes the defining representation $K^n$ of $M_n(K)$? If not, what can be said about finitely generated projective modules over these rings and their relationship to irreducible representations of $G$ over fields?

This is an abstract way of phrasing a more concrete question about whether every irreducible representation of $G$ admits a $G$-invariant lattice over $\mathbb{Z}[\zeta_{|G|}][|G|^{-1}]$ or something like that.
Edit: Some partial progress towards Q1. Consider the universal splitting: there is a functor sending a commutative ring $R$ to the set of splittings over $R$, namely the set of isomorphisms between $R[G]$ and $\prod_n M_{n_i}(R)$ over $R$, where $n_i$ are the dimensions of the irreducible representations of $G$ (over $\mathbb{C}$, say). This functor is representable by a finitely presented ring, since specifying a splitting amounts to specifying two $R$-linear maps going in each direction (which amounts to two $|G| \times |G|$ matrices) together with the conditions that they are homomorphisms and also that they are mutually inverse, which are polynomial conditions on the entries of the corresponding matrices. So if we take $R$ to be this representing ring $R_{\text{Iso}}$ then not only does $G$ admit a splitting over $R_{\text{Iso}}$ but a splitting over a ring is precisely a ring together with a homomorphism from $R_{\text{Iso}}$.
We know that $R_{\text{Iso}}$ is not the zero ring because it admits homomorphisms to $\overline{\mathbb{Q}}$ (given by splittings over $\overline{\mathbb{Q}}$). But furthermore we know it admits homomorphisms to $\overline{\mathbb{F}_p}$ for all primes $p$ not dividing $|G|$. This tells us that all such primes $p$ are non-invertible in $R_{\text{Iso}}$, which is promising, and in some sense already answers the motivating question (since the irreducible representations of $G$ over $\mathbb{C}$ are defined over $R_{\text{Iso}}$ and this ring admits a homomorphism to $\mathbb{C}$ and to $\overline{\mathbb{F}_p}$ for all $p \nmid |G|$). But it would be nice to find a quotient of $R_{\text{Iso}}$ which is a ring of $S$-integers and which still only inverts primes dividing $|G|$.
 A: Here is a sketch of an argument.
Let $Irr(G)=\{\sigma\colon G\to GL(V_\sigma)\}$ denote the set of irreducible representations of $G$ over $\mathbb C$. Suppose $R\subset\mathbb C$ is a ring such that any $\sigma\in Irr(G)$ can be defined over $R$, i.e., there is a representation $\tilde\sigma\colon G\to GL(\tilde V_\sigma)$ for some free $R$-module $\tilde V_\sigma$ such that the induced representation $\tilde V_\sigma\otimes_R\mathbb C$ is isomorphic to $\sigma$.
Then, there is a ring homomorphism
$$\begin{align*}
\phi\colon R[G]&\to \prod_{\sigma\in Irr(G)}End_R(\tilde V_\sigma)\\
g\in G&\mapsto (\sigma(g))_{\sigma\in Irr(G)},
\end{align*}$$
which is injective since it is injective after tensoring with $\mathbb C$. Moreover, the cokernel must consist of torsion.
In particular, if $G$ has exponent $m$, then every irreducible representation of $\mathbb C$ can be realized over $\mathbb Q(\zeta_m)$ by Serre's Linear Representations of Finite Groups Theorem 24. Thus, in particular, they can be realized over the ring of integers $\mathbb Z[\zeta_m]$. The cokernel of $\phi$ consists of torsion, so localizing sufficiently kills it, and $\phi$ becomes an isomporphism.
