= 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.
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