How do you visualize polynomials rings and their quotient? When working in the framework of group theory I could easily visualize the group by imagining its elements and I could see the group "collapse" when forming the quotient group by a normal subgroup by simply imagining the elements of the subgroup all being collapsed onto the identity, same thing for the other cosets.
But for polynomial rings I have a very hard thing visualizing this and it has lead to a poor understanding of ideals and quotient rings since I cannot rely on my visualize intuition anymore. I cannot make sense of $\mathbb{R}[x]$, its ideal $x^2+1$ and its quotient ring by that ideal.
I was wondering if you had any help in this regard because I am struggling here.
 A: I visualize elements of the polynomial ring $\mathbb R[x]$ as, well, polynomials, although perhaps it helps to keep one's focus on the $0$ degree term:
\begin{align*}
    & a_0 \\
a_1 x + & a_0 \\
a_2 x^2 + a_1 x + &a_0 \\
a_3 x^3 + a_2 x^2 + a_1 x + &a_0
\end{align*}
and so on.
And then, when you mod out by $x^2+1$, welp, that's just substituting every $x^2$ by $-1$, and every $x^3$ by $-x$,
\begin{align*}
    & a_0 \\
a_1 x + & a_0 \\
a_2 (-1) + a_1 x + &a_0 \\
a_3 (-x) + a_2 (-1) + a_1 x + &a_0
\end{align*}
and so on... and then you have to collect terms of course, which shifts the terms formerly known as the $x^3$ and $x^2$ terms two to the right, and shifts the terms formerly known as the $x^5$ and $x^4$ terms four to the right, and so on.
A: It's the same. Literally. Ideals of a ring are normal subgroups of their additive group, and the quotient of the ring by the ideal is just the quotient of the additive group by this normal subgroup. And then the multiplicative structure can be added on top. At their core, rings are groups with additional structure, and their quotients reflect this.
