characterization of quotient rings In the ring $\mathbb{Q}{[x]}$ consider the following subset
$I = \{f(x) (x^2 + 3) | f(x) \in \mathbb{Q}[x]\}$
Prove that $I$ is an ideal in $\mathbb{Q}[x]$ and express the quotient ring $\mathbb{Q}[x]/I$.
First I'm not sure of all the conditions to verify to be sure in an ideal.
I verified $I$ is an additive subgroup since $f(x)(x^2 + 3) + g(x)(x^2 + 3) = h(x) (x^2 + 3)$ with $h = g + f$.
Then it is also a two-sided ideal $g(x) \cdot f(x)(x^2 + 3)\in I$ with $g \in \mathbb{Q}{[x]}, f(x)(x^2 + 3) \in I$ and $f(x)(x^2 + 3) \cdot g(x) \in I$ as well.
Is it enough, or are there other conditions to be satisfy?
Regarding the characterization of $\mathbb{Q}[x]/I$, I'm uncertain. This should be made of all congruence classes with regards to the equivalence relation ~ defined as:
$g \sim f \quad \iff f - g \in I \quad for \quad g,f \in \mathbb{Q}[x]$
my manual express it as $R/I := \{a+I | a \in \mathbb{Q}[x]\}$
with
$a+I := \{a + i | i \in I \} = \{x \in \mathbb{Q}[x] | x \sim a\}$
By the division algorithm, for every $p(x) \in \mathbb Q[x]$ there exists $q(x),r(x) \in \mathbb Q[x]$ such that $$p(x) = (x^2+3)q(x)+r(x)$$ and that $\deg r(x) &lt; \deg(x^2+3) = 2$ whenever $r(x) \neq 0$; put in another way: for every $p(x) \in \mathbb Q[x]$ there exists $q(x) \in \mathbb Q[x]$ and $a,b \in \mathbb Q$ such that $p(x) = (x^2+3)q(x)+(a+bx)$, which is the same as saying $$p(x)+I = (a+bx)+I.$$ Thus $\mathbb Q[x]/I = \{(a+bx)+I : a,b \in \mathbb Q\}$.
By the division algorithm, for every $p(x) \in \mathbb Q[x]$ there exists $q(x),r(x) \in \mathbb Q[x]$ such that $$p(x) = (x^2+3)q(x)+r(x)$$ and that $\deg r(x) &lt; \deg(x^2+3) = 2$ whenever $r(x) \neq 0$; put in another way: for every $p(x) \in \mathbb Q[x]$ there exists $q(x) \in \mathbb Q[x]$ and $a,b \in \mathbb Q$ such that $p(x) = (x^2+3)q(x)+(a+bx)$, which is the same as saying $$p(x)+I = (a+bx)+I.$$ Thus $\mathbb Q[x]/I = \{(a+bx)+I : a,b \in \mathbb Q\}$.