To prove ; $pa:=a+a+... p $ times $=0 , \forall a \in R$ and $a^p=a , \forall a \in R$ for some prime $p$ then the ring $R$ is commutative

Question

If in a ring $R$ , $\exists $ prime $p$ such that $pa:=a+a+... p $ times $=0 , \forall a \in R$ and $a^p=a , \forall a \in R$ , then how to prove that $R$ is commutative ? I would not want to use something big like Jacobson's theorem concerning $a^{n(a)}=a$ , it would be overkill . Please help