Abstract
A ring R is semiprime (semiprimitive) if and only the intersection of the prime (primitive) ideals is zero. Then, R is a subdirect product of prime (primitive) rings 26.6 and 26.13. The (McCoy) prime radical of a ring is defined to be the intersection of the prime ideals, and is characterized as the set of all strongly nilpotent elements of R (theorem of Levitzki 26.5). When R is commutative, this is just the set of nilpotent elements.
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