Let $R$ be a commutative ring with unity. Let $I$ be an ideal of $R.$ Then I know that $$\operatorname {rad} ({I}) = \bigcap\limits_{\substack {p \supseteq I \\ \text {p prime}}} p.$$
Now let $I = (0).$ Then by the similar reasoning $$\operatorname {rad} ({0}) = \bigcap\limits_{\text {p prime}} p.$$ Now since every prime ideal $p \supseteq (0)$ and $(0)$ is itself a prime ideal in an integral domain so I think if $R$ is taken to be an integral domain then $\operatorname {rad} ({0}) = (0).$ Though it is intuitively clear because integral domains cannot have any non-zero nilpotent element i.e. every integral domain is a reduced ring and hence $\operatorname {rad} {(0)} = (0).$
Is my reasoning correct at all? Would someone please verify it?
Thank you very much.