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# prime radical

The prime radical or lower nilradical of a ring $R$ is the intersection of all the prime ideals of $R$.

Note that the prime radical is the smallest semiprime ideal of $R$, and that $R$ is a semiprime ring if and only if its prime radical is the zero ideal.

Related:

RadicalOfAnIdeal, Nilradical

Synonym:

lower nilradical

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

16N80*no label found*

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