Consider an abelian algebra, $R$, over the field $K$ with the properties that every residue field of $R$ is (canonically) isomorphic to $K$ (I'm not sure but I think this is necessary, otherwise we could be talking about $\mathbb{R}[x]$ as a $\mathbb{Q}$-algebra) and that for every maximal ideal $m$, $R/m^2\cong K\oplus m/m^2$ as additive groups (and let $\phi:R\rightarrow R/m^2$ be the natural quotient map). For every $s \in R$ call the $K$ factor of $\phi(s)$ the 'value of $s$ at $m$' and call the $m/m^2$ factor 'the derivative of $s$ at $m$'. Under what conditions does 'the derivative of $s$ vanishes at each $m$' imply that 'the value of $s$ is the same at every $m$'? What are the conditions if we talk about jet spaces instead of tangent spaces (i.e. replace every instance of $m^2$ with $m^k$ for a specific or arbitrarily large $k$)?
There are obvious examples and counterexamples. The ring of polynomials over a field of characteristic 0 satisfies this condition, as does the ring of smooth functions on a connected manifold (and maybe even $p$ times differentiable functions, even though the Zariski tangent space isn't the normal tangent space). Any ring of functions on a disconnected space obviously doesn't satisfy this, but there are also connected counterexamples, such as the ring of polynomials over a field of positive characteristic, although in that case talking about arbitrarily large jet spaces gives you an analogous statement (i.e. if every jet of a polynomial is constant, then the polynomial is constant).
