first of all, this claim that "the Jacobson radical contains all nilpotent elements" is true in commutative rings not in all rings. for example but obviously has non-zero nilpotent elements.

what is true in general is that the Jacobson radical contains all nilpotent ideals.

to answer your question, just use this fact that if and only if is invertible for all you'll eventually need to consider two cases: and