$R[x]$ is semiprimitive if and only if $R$ has no non-zero nilpotent ideals

Question

Here is the question I'm working on: Show $R[x]$ is semiprimitive if and only if $R$ has no non-zero nilpotent ideals.

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Here's my attempt at $semiprimitive \Rightarrow no\space non-zero\space nilpotent\space ideal$:

If $R[x]$ semiprimitive, then $J=(0)$, but we've already shown that every nilpotent element is in $J$, so there are no nilpotent ideals.

($J$ here denotes the Jacobson Radical).

Is this correct? How would I go about proving the other direction?