I'm struggling with the following question:
Find an example of a proper ideal in the real-valued polynomial ring $\mathbb{R}[x]$ which is not maximal, but is co-maximal with the ideal $(x^2)$.
I know that the ideals $I$ and $J$ in a ring $R$ are co-maximal if $I+J = R$. Also, I know that if an ideal contains a unit, say $1$, then it must be equal to the entire ring.
So, by putting $I = (x^2) I$ need to find an ideal $J$ which satisfies the aforementioned conditions s.t. $1$ is expressible as a linear combination of elements of $I$ and $J$.
Am I on the right track? I guess my main issue is understanding which elements are maximal in the polynomial ring and which aren't.