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How to show that $\mathbb Z+x \mathbb Q[x]$ is a Bezout domain, that is, the sum of two principal ideals is again a principal ideal ? Or at least, how to show that it is a GCD domain ? (This will then provide an example to Looking for an example of a GCD domain which is not a UFD.)

Please help. Thanks in advance.

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  • $\begingroup$ See this answer for a proof that $\mathbb Z+X\mathbb Q[X]$ is not a UFD, and this Wiki page for a proof that it is a Bezout domain. $\endgroup$ – user26857 May 4 '15 at 10:43
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Hints: It suffices to prove that every two generated ideal is principal. Let $R:=\mathbb Z+x \mathbb Q[x]$. Note that a polynomial $f$ in $R$ can be written as product irreducible elements iff the constant term of $g$ is not zero (equivalently, lies in $x\mathbb Q[x]$). So it can be divided to two cases: at least one of the generators is outside $x\mathbb Q[x]$; both generators lie in $x\mathbb Q[x]$. And the following properties may be useful: if $p$ is a prime number, then $R/(p)$ is a field; and if $g$ is a irreducible polynomial in $\mathbb Q[x]$ of which the constant term is $1$, then $R/(g)$ is a field.

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