 Finite minimal primes and nilpotent prime radical

In a left Noetherian ring, Noetherian induction shows that every proper two-sided ideal contains a finite product of prime ideals of a noncommutative ring that contain it. A maximal counterexample would be nonprime and contain a product of two strictly larger ideals; their prime products give the contradiction. Every prime over the original ideal contains a factor, so the inclusion-minimal factors give all its minimal primes over an ideal. Replacing each factor by a minimal prime beneath it gives a product of minimal primes inside the original ideal, with repetitions allowed. Applying this to zero makes a power of the prime radical zero.