If $I[x]$ is a radical ideal of $R[x]$, then show that $I$ is a radical ideal of $R.$

Question

Show that $I$ is a radical ideal of a commutative ring $R$ iff $I[x]$ is a radical ideal of $R[x].$

The problem is in the ''only if'' part.If $I[x]$ is a radical ideal of $R[x],$ then $I[x] = \sqrt J$ for some ideal $J$ of $R[x].$

Then, how to find an ideal of $R$ whose radix will be $I?$ Will such an ideal be unique?

Answer 1

Note that $I \subset R$ is radical if and only if the quotient ring $R/I$ is reduced (has no non-trivial nilpotents). Your statement is now immediate since $R/I \cong R[x]/I[x].$