Nilradical is the intersection of finitely many minimal prime ideals.

Let $$R \neq \{0 \}$$ be a commutative ring with identity. Suppose that $$R$$ has only finitely many minimal prime ideals $$p_1,\dots , p_s.$$ Then $$\sqrt {0} = \bigcap\limits_{i=1}^{s} p_i.$$

I know that $$\sqrt 0 = \bigcap\limits_{p\text { prime}} p.$$ But from here how do I prove the above theorem? Would anybody please help me regarding this.

Thank you very much.

Every prime ideal contains a minimal prime ideal, so the intersection of all prime ideals is the same as the intersection of all minimal prime ideals. Given that these are $$p_1$$, $$p_2$$, ..., $$p_s$$, it follows that $$\sqrt 0 = \bigcap\limits_{\text {p prime}} p = \bigcap\limits_{i=1}^{s} p_i.$$