Description of the elements of (p(x))\(q(x))

Question

I am studying Rings and i saw a description of the elements of $R[x]/I$ where $I=Rf=(f(x))=\{p(x)\in R[x]: p(x)=f(x)q(x) \text{ for some $q(x)$ } \}$ (suppose that $deg(f)=n$) then $$R(x)/I=\{c_0+c_1x+\cdots+c_{n-1}x^{n-1}+I:c_1,\ldots,c_{n-1}\in R \}$$ This makes sense because every polynomial is equivalent with its quotent when divided with $f(x)$.
Now lets say we want to find all the ideals of $\mathbb{R}[x]/(x^4+x^2)$. Using the the fourth isomorphism theorem (Correspondence theorem) we can see that all the ideals of $F[x]/(p(x))$ (F=field) are in the form of $(g(x))/(f(x))$ for $g(x)|f(x)$. For example one of the ideals of $\mathbb{R}[x]/(x^4+x^2)$ is $(x^2+1)/(x^4+x^2)$ because $x^2+1|x^4+x^2$. Since we have a clear descreption of the classes in $R(x)/(f(x))$ my question is how could one describe in an intuitive way the elements in $(x^2+1)/(x^4+x^2)$ and more generally the elements in $(p(x))/(q(x))$ where $p,q\in F[x], F=\text{field} $ and $p|q$. Thanks in advance

Answer 1

I'm not sure how intuitive it is, but one way of thinking about the elements in $\overline{(p(x))}$ (the image of the ideal $(p(x))$ in the quotient) is to just write out what these elements are in the original ring and use the ideal $(q(x))$ to simplify things. In your example of $\overline{(x^2+1)}\subset\mathbb{R}[x]/(x^4+x^2)$, the elements in the original ideal $(x^2+1)$ look like $a_0(x^2+1)+a_1x(x^2+1)+a_2x^2(x^2+1)+\dots$ with the $a_i\in\mathbb{R}$. However, when you quotient out by $(x^4+x^2)$, the terms which are multiples of $x^2(x^2+1) = x^4+x^2$ will vanish, so you can simplify the expression above to $a_0(x^2+1)+a_1x(x^2+1)$. These will be elements in the image of your ideal after quotienting.