$p$-Sylow subring
Solution 1:
For $r \in R$ fixed, the maps $s \mapsto rs$ and $s \mapsto sr$ are endomorphisms of the abelian group structure on $R$. Now use the fact that for an abelian group $G$ with Sylow subgroup $P$, any endomorphism $\phi$ of $G$ stabilizes $P$: $\phi(P) \subseteq P$ (this follows from standard Sylow theory). Therefore the Sylow subgroups are actually (two sided) ideals in $R$.