The question's in the title. For instance, if $R$ contains $2$ then there are an infinite number of pairs of prime principal ideals $(p),(q)$ such that $p = q + 2$. I just made that up and it's probably not true. But to give you an idea....
2026-03-25 11:06:44.1774436804
Is there a generalization of the twin prime conjecture to rings or certain rings?
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In $\Bbb Z[x]$, both $(x+k)$ and $(x+k+2)$ are prime for every integer $k$.