I am doing this exercise about primary ideals and need some help.
Let $R$ be an integral domain and $\varphi: R \hookrightarrow R[x]$ is the canonical embedding. Assume that $I$ is a primary ideal in $R$. Show that $I^e$ is also primary in $R[x]$.
My approach is showing that $I^e \neq R[x]$ and $\{\text{zero divisors of } R[x]/I^e\} \subseteq {\rm nil}(R[x]/I^e)$. Observe that $I^e=I[x]$ (the polynomial ring over $I$), therefore $R[x]/I^e=R[x]/I[x] \cong (R/I)[x]$. The main thing to do is proving the inclusion. Basically I try to write down everything based on definitions, but it seems to be lengthy. So I write here to discover some more effective ways to nail this.
Any help will be appreciated.
