This question/problem really comes from a fact in algebraic geometry, where it says that given an irreducible variety $V$ ($\dim V \geq 2$) then for any given pair of points $x,y\in V$ there is an irreducible curve $C$ connecting them.

This can proved by invoking Bertini's theorem [See Lazarsfeld R. - Positivity in Algebraic Geometry 1, example 3.3.5].

Translating this back to coordinate rings we get something like: Given a finitely generated $k$-algebra $R$ which is a domain, $k=\bar{k}$ (alg.closed) then for any pair of maximal ideals $m_1,m_2\subset R$ there is a prime ideal $p\subset m_1\cap m_2 $ with $\dim R/p =1$.

So my question is therefore: Is there a purely ring theoretical argument for this fact? In what generality does it hold?

The reason why i emphasize on purely is because, and i am no expert on this, but apparently the Bertini type argument can be (re-)formulated algebraically, however this, I have been informed, will not be pretty...