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Is it true that if $R$ is a PID with infinitely many maximal ideals, then every element of $R[x]$ of degree $n\ge1$ is a sum of two irreducible polynomials in $R[x]$? Even if this is not true in general, then is it at least true for $R=\mathbb Z$? And if it is true, then is it true when $R$ is a Noetherian domain (i.e. every ideal is finitely generated) with infinitely many maximal ideals?

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Please see "On polynomial rings with a Goldbach property" from Paul Pollack (Wayback Machine) for instance.

You have the theorem $1$ : if $R$ is a noetherian integral domain and has infinitely maximal ideals, then "Goldbach" is true. Note that the theorem $1$ is consequence of the theorem $5$ he proves in page $3$.

Removing the noetherian hypothesis is not possible, he gives in the introduction the counter-example where $R$ is the ring of all algebraic integers (complex number that are solution of unitary polynomial with coefficients in $\mathbf{Z}$). Over $R$, there are no irreducible polynomials of degree $n > 1$, as every non constant polynomial in $R[T]$ can be written as a product of linear factors. But you have infinitely many maximal ideals in $R$ : for every positive prime $p$ in $\mathbf{Z}$, by the Zorn’s lemma, you have a maximal ideal of $R$ containing $(p)$ and distinct primes $p$ correspond to distinct maximal ideals... By the way, you also cannot remove the "infinitely many maximal ideals" hypothesis, as he points after.

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