I have a proposition which says that an ideal $I$ is a prime ideal in $R$ iff its quotient ring $R/I$ is a domain. I know that for an ideal to be a prime ideal, $xy\in I\implies x\in I\lor y\in I$. My thought is this: $x\notin \langle x^2-x\rangle$ and $x-1\notin \langle x^2-x\rangle$, but $x(x-1)=x^2-x\in \langle x^2-x\rangle$, so $\mathbb{Q}[x]/\langle x^2-x\rangle$ is not a domain. Am I thinking this right?
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1The idea is correct. – egreg Jan 01 '21 at 17:12
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@egreg Ok, cool. Is the execution wrong, though? – infiniteMonkeyCage Jan 01 '21 at 17:13
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Not wrong, but well known and often asked again. $x^2-x=x(x-1)$ is reducible. – Dietrich Burde Jan 01 '21 at 17:16
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@KasperLarsen You just have to dress it up: proving the statement $x\notin(x^2-x)$, for instance (which is easy, but is necessary nonetheless). – egreg Jan 01 '21 at 17:25
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@egreg I see, thank you! – infiniteMonkeyCage Jan 02 '21 at 07:43