I have an exercise to show that $\mathbb{Q}S_3 \cong \mathbb{Q} \oplus\ \mathbb{Q}\ \oplus\ M_2(\mathbb{Q}) $ , where $M_2(\mathbb{Q})$ is ring of $2$ by $2$ rational matrices and $\mathbb{Q}S_3$ is group ring.
How can I establish ring isomorphism between them. After several attempts, I cannot find a suitable isomorphism from $\mathbb{Q}S_3 \to \mathbb{Q} \oplus\ \mathbb{Q}\ \oplus\ M_2(\mathbb{Q})$.
First let's treat the case of $\mathbb{C}[G]$. for any finite group $G$, $\mathbb{C}[G]\simeq \prod_i Hom_\mathbb{C}(V_i)$ where the $(V_i,\rho_i)$ are the non-isomorphic irreducible representations of $G$, and the isomorphism is given by $g\mapsto (\rho_i(g))_i$.
Now if $G=S_3$, this gives $\mathbb{C}[S_3] \simeq M_2(\mathbb{C})\times \mathbb{C}\times \mathbb{C}$ since $S_3$ has $3$ irreducible representations : the standard one, the trivial one, and the signature. But you may check that they all are defined over $\mathbb{Q}$, meaning that $\rho(g)\in M_r(\mathbb{Q})$ for all these representations.
So this gives you by restriction $\mathbb{Q}[S_3] \simeq M_2(\mathbb{Q})\times \mathbb{Q}\times \mathbb{Q}$, and the isomorphism is $g\mapsto (\rho(g),1,\varepsilon(g))$ where $\rho:S_3\to M_2(\mathbb{Q})$ is the standard representation and $\varepsilon(g)$ is the signature.