Consider a polynomial of degree at least 2 with all coefficients in $\mathbb{R}$. We are concern with set of solution for the polynomial in $End(\mathbb{R},+)$ - the endomorphism ring of the abelian group $\mathbb{R}$ under addition. The question is: is it possible for the number of solution to be finite?
My hunch is no, due to the polynomial lacking control over transcendental number. This at first appeared to be a simple application of Zorn's lemma, but it turns out to be not straightforward: one particular difficulty I'm having is whether there exist a proper invariant subspace for a particular solution. I'm sure there would be question like this somewhere, but I don't even know what to search for to find it.
Thank you for your help.
Edit: It occured to me some time after the original answer that you're going to want the operator assigned to the formal variable to commute with the ''multiplication by r" $M_r$ operators. But then everything is determined by where 1 is sent, and so the operators which commute with the $M_r$ are just the $M_r$ themselves, and so the solutions to any particular polynomial are just multiplication by the usual real solutions.
However, if you want some kind of ring consisting of sums of formal words in some variable $X$ and real numbers, some of what I said below still holds and I'll keep it around.
As Asaf points out, $G := (\mathbb{R},+)$ as a $\mathbb{Q}$-vector space is isomorphic to $FVS_{\mathbb{Q}}(X)$, where $X$ has cardinality $2^{\aleph_0}$ and $FVS_k$ is the free vector space over $k$ functor. Since $X \cong \bigsqcup_{n \in \mathbb{N}} X_n$ with $X_n$ also of cardinality $2^{ \aleph_0}$ and $FVS$ takes disjoint unions to direct sums, we have that as a $\mathbb{Q}$-vector space, $G \cong \bigoplus_{n \in \mathbb{N}} G$, and this direct product decomposition also works as $\mathbb{Z}$-modules.
Now, let $f(X) \in \mathbb{R}[X]$. If $f$ has more than one solution over the endomorphism ring of $G$, say $T_0,T_1$, then for each function $\phi:\mathbb{N} \rightarrow \{0,1\}$, we can construct using the direct sum decomposition the endomorphism $\bigoplus_{n \in \mathbb{N}} T_{\phi(n)}:\bigoplus_{n} G \rightarrow \bigoplus_n G$. So if a polynomial has more than one solution, it has to have infinitely many.