Abstract Algebra Math notation, kernel

Question

So I defined

$$\theta:R \rightarrow R\backslash I$$

by $$\theta(x) = [x]_I$$ and

$$\phi: R[x]\rightarrow (R\backslash I)[x]$$ by $$\phi(a_nx^n +\dotsb+a_1x+a_0) = \theta(a_n)x^n+\dotsb+\theta(a_1)x+\theta(a_0)$$

I know that $\ker\phi= I[x]$ but I can't express it in math notation. :'(

Here's an attempt: Let $f(x)=a_nx^n +.....+a_1x+a_0$

$\ker\phi = \{f(x) \in R[x]|\phi(f(x)) = 0\} =\{f(x) \in R[x]|\theta(a_i) = 0 \forall i\in R\} \rightarrow$ {polynomials with coefficients in I}$=I[x]$ How can I express this? Thanks :)

Answer 1

$\theta(a_i) = 0$ is equivalent to $a_i \in I$. So you get $\{ a_n x^n + \dots + a_0 \in R[x] \mid \theta(a_i) = 0 \text{ for all $i$} \} =$ $\{ a_n x^n + \dots + a_0 \in R[x] \mid a_i \in I \text{ for all $i$} \} =$ $I[x]$.