What is this ideal equal to? What is it called? "composition ideal in $R[X]$"

Question

Let $R$ be a ring and $f(X)=f_0+f_1X+\dots +f_n X^n\in R[X]$. Define $f(J) \equiv f_0 + f_1 J + \dots + f_n J^n$ where $J^k$ is the $k$th power ideal, and $A + B = \{a + b : a \in A, b \in B\}$.

Let $J, K$ be two ideals of $R$. Let $I = \{ f \in R[X]: f(J) \cap K \neq \varnothing\}$. Then $I$ is an ideal of $R[X]$.

Answer 1

Note that $X\in I$, since $0\in J\cap K$. It follows that $f\in I$ iff $f_0\in I$ (since $f-f_0$ is divisible by $X$). A constant $f_0$ is in $I$ iff $f_0\in J\cap K$. So $I$ consists of all the polynomials whose constant term is in $J\cap K$.