Short exact sequence of $\mathbb{R}[X]$-modules that does not split

Question

What is an example of a short exact sequence of $\mathbb{R}[X]$-modules that does not split?

Answer 1

Here are the basic relevant facts to get you thinking about the possibilities:

A short exact sequence $$0 \to A \overset{f}{\to} M \overset{g}{\to} B \to 0$$ is the data of a submodule $f(A) \subset M$, where $f$ encodes $A \simeq f(A)$, and its quotient $M/f(A) \simeq B$ via the quotient map $g$. The sequence is said to split if $M$ has a complementary submodule to $f(A)$ isomorphic to $B$, which will make the sequence isomorphic to the most basic possible example of a short exact sequence, $$0 \to A \overset{i}{\to} A \oplus B \overset{\pi}{\to} B \to 0,$$ where $i$ is inclusion into the first coordinate in $A \oplus B$ and $\pi$ is projection onto the second coordinate of $A \oplus B$.

An $\mathbb R[X]$-module is the same thing as the data of an $\mathbb R$-vector space $V$ together with a linear transformation $X: V \to V$ encoded by multiplication by $X$. A submodule corresponds to an invariant subspace of this linear transformation. So what you need is a linear transformation together with an invariant subspace with no complementary invariant subspace. This is a basic linear algebra problem.