Is it possible to show whether or not $ \{xf(x)+3g(x)\;|\;f(x),g(x) \in \mathbb{Q}[x]\} $ is an ideal (or main ideal in $\mathbb{Q}[x]$)?
I know how to prove it for $\mathbb{Z}[x]$, but what with $\mathbb{Q}[x]$?
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3Do you mean "principal" ideal = an ideal generated by onesingle element? And should it be $;xf(x)+3g(x);,;;f,g\in\Bbb Q[x];$ ? – Timbuc Jan 01 '15 at 16:17
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Note that $3$ is invertible in $\mathbb{Q}[X]$. There is thus a very simple description for your set, which will allow to answer the question directly.
quid
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The ring $\;\Bbb F[x]\;$ is a principal ideal ring (PID) iff $\;\Bbb F\;$ is a field, so any ideal you find in $\;\Bbb Q[x]\;$ is principal.
OTOH, $\;\Bbb Z[x]\;$ is obviously not principal (since $\;\Bbb Z\;$ is not a field), so I'm not sure what you meant by " knowing to prove it in $\;\Bbb Z[x]\;$" .
Timbuc
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1See this answer for a way to prove this using only high-school algebra (evaluation of polynomials). – Bill Dubuque Jan 01 '15 at 16:42