Let $k$ be an algebraically closed field and $R$ the polynomial ring in $n$ variables over $k$. If $J$ is an irreducible ideal of $R$ then it is a prime ideal as well.
To establish this statement I have succeeded to show that J would be primary and it will follow if we can show that it is radical as well... but I am not seeing anything to establish it even using Nullstellensatz... any help is appreciated.
You cannot "establish this statement" because it is false!
For example $(X^2)\subset k[X]$ is irreducible but not prime.
Remark The correct implications (valid in any noetherian ring) are:$$ \text {prime} \implies\text {irreducible} \implies \text {primary} $$