Monic polynomials $f\in R[x]$ with $f-a$ irreducible for almost all values of $a\in R$

Question

I assume all the rings to be commutative and unital. Recall that an element $b$ of a ring $S$ is irreducible if $b\notin\{0\}\cup S^*$ and $b$ is not the product of two noninvertible elements in $S$.

Let $R$ be a ring. The polynomial $f=x$ satisfies that $f-a$ is irreducible for all $a\in R$. For general $f$ with $\operatorname{deg}(f)>1$, we have $f-f(0)=xg$, with $g\in R[x]$ monic and nonconstant, hence noninvertible.

Is there some example of a ring $R$ and a monic polynomial $f\in R[x]$ and $\operatorname{deg}(f)>1$, such that $f-a$ is irreducible (in $R[x]$) for all $a\neq f(0)$?

Answer 1

Yes. Consider the ring $R = \mathbb{F}_2[T]$ and $f = T^3 + T + 1 \in R$. Then f is irreducible, since it has no roots in $\mathbb{F}_2$ and is of degree $\text{deg}(f) = 3$. Now $f(0) = 1$, which means that $a = 0$ is the only other element to consider. Since $f = f + 0$ is irreducible, it is irreducible for all $a \neq f(0) = 1$.

Answer 2

Yes. consider an Artin–Schreier polynomial $$f(x)=x^p - x - \alpha$$ in $\mathbb{F}_p[x]$, where $\alpha\neq 0$. Then $f(x)+\beta$ is irreducible for all $\beta\neq \alpha$ in $\mathbb{F}_p$.