I know that a polynomial ring over a field is a PID, does this property also hold for a polynomial ring over a skew field? Is there maybe something else that characterise the ideals in that ring ?
2026-03-29 00:06:40.1774742800
Ideals in a polynomial ring over a skew field
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The argument for $F[x]$ being a PID for a field F carries over to the case with F a division ring.
Given a right ideal $T$, pick a monic elements minimal degree $g$ in $T$. Given a $y\in T$, you can still use a division algorithm (bring careful to multiply on the right) to find q,r such that $y=gq+r$ with the degree of r lower than g. Then you conclude r=0 and see that g generates $T$.
The left ideals are principal as well, and so are the ideals, being special cases.