Radical of an ideal in $R [x]$

Question

Let $\frak {I}$ be an ideal of $R[x]$, the polynomial ring over a commutative ring with identity $R$. Is it true that the radical of $\frak{I}$, the intersection of all prime ideals containing $\frak{I}$, is equal to the radical of a homogeneous ideal of $R[x]$, an ideal generated by a set of homogeneous elements?

Answer 1

Certainly not. For instance, if $R$ is reduced, then the ideal $(x-1)$ is radical (since $R[x]/(x-1)\cong R$ by the evaluation homomorphism sending $x$ to $1$), and the only homogeneous element it contains is $0$ (since a homogeneous element $ax^n$ maps to $a$ under the above evaluation homomorphism).