Prime radical

Remarks

Example: let $R$ be the ring of upper triangular $2 \times 2$ matrices over a field $k$ and $N$ its strictly upper triangular matrices. Then $R/N \cong k \times k$, the prime ideals of $R$ are the kernels of the two maps $R \to k$ that take a diagonal entry, and their intersection is $N$. Non-example: the matrix unit $e_{11}$ is not in the prime radical of $R$, since it is a nonzero idempotent, while every element of the prime radical is nilpotent. Equivalent forms: $\operatorname{Nil}_*(R)$ is the smallest semiprime ideal of $R$, and it is the set of strongly nilpotent elements of $R$. It is also called the lower nilradical or the Baer radical, and it is contained in the upper nilradical. For a commutative ring it equals the nilradical.