Radical of an ideal

Remarks

The set $\sqrt{I}$ is an ideal of $R$ that contains $I$: if $a^m \in I$ and $b^n \in I$, then every term of the binomial expansion of $(a + b)^{m + n - 1}$ lies in $I$. Example: in $\Z$, $\sqrt{12\Z} = 6\Z$, since $6^2 = 36 \in 12\Z$, and if $r^n \in 12\Z$ then the primes $2$ and $3$ divide $r$. Non-example: $4 \notin \sqrt{12\Z}$, because no power of $4$ is divisible by $3$. The radical of the zero ideal is the nilradical of $R$. An ideal equal to its radical is a radical ideal. Other notations are $\operatorname{rad}(I)$ and $r(I)$.