I would be shocked if the following were not true, but I can't seem to see a proof.


Claim:


Let $R$ be an integral domain containing an AC (uncountable if you wish) field $k$. Let $a, b \in R$, and suppose that the ideal $(a, b)$ is height 2. 


Then for general $\alpha, \beta \in k$, the element $\alpha a + \beta b$ is irreducible. 


Thanks!


Sue 



irreducibility of generic linear combination of polynomials?
This is not true even over $\mathbb C$. Take $\mathbb C[x,y]$ and $x^2, y^2$. You need general combination of a regular sequence of length at least $3$. Search for "local Bertini theorem" and "Flenner". 


ADDED: the relevant reference is Satz 4.9 and 4.10 (*Die Sätze von Bertini für lokale Ringe* by H. Flenner, Mathematische Annalen, (299), 1977). This works for $n$ elements such that the ideal generated by them has height at least $3$ (over a infinite field of char. $0$). There is no hope in char. $p>0$ no matters how many elements you pick, since one can expand Karl's example. 


The height at least $3$ condition can't be weakened (think about $x^2, xy, y^2$ over $\mathbb C$). However, in the case of $2$ elements, if you assume $a,b$ are irreducible to begin with, then I would guess what you want has a much better chance. 


