i) Show that any element of $R$ is congruent modulo $I$ to a unique polynomial of the form $ax+b$ where $a,b \in \mathbb{Q}$?
ii) Show that any element of the quotient ring $R/I$ is of the form $\widehat{ax+b}$ for some $a,b \in \mathbb{Q}$?
I've attempted both questions.
For i) I use the division algorithm to show that for each polynomial $f(x) \in R$, there are unique polynomials $r(x), q(x) \in R$ such that
$$f(x) = q(x)(x^2+2x+2) + r(x)$$
where $r(x) = ax + b$ and $a,b \in \mathbb{Q}$. Therefore $f(x) - r(x) \in I$ so $f(x)$ is congruent to $r(x)$ modulo $I$.
As for ii) I wrote that $\widehat{f(x)} = \widehat{r(x)}$ because the difference of the two coset representitives lies in $I$. Therefore, $\widehat{f(x)} = \widehat{r(x)} = \widehat{ax+b} \in R/I$.
Am I correct when answering both questions or have I left anything out?
Thank you
