Prove that every ideal of $R=\mathbb{Z}[x]$ can be generated by at most two elements of $R$.
$\mathbb{Z}[x]$ is polynomial ring over $\mathbb{Z}$.
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Have you tried proving it in a more specific case, like for instance quadratic polynomials? – Robert Soupe Nov 12 '19 at 17:21
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1You added the [tag:model-theory] tag. Are you looking for model-theoretic proof or something? Or did you mean “module theory”? – k.stm Nov 12 '19 at 17:22
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5What about $\left<X^2,2X,4\right>$? – Angina Seng Nov 12 '19 at 17:24
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5If you meant to say "every prime ideal", see https://math.stackexchange.com/questions/3317685/every-prime-ideal-in-mathbbzx-is-generated-by-at-most-two-elements?rq=1 – Nate Eldredge Nov 12 '19 at 17:37
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1See also: https://math.stackexchange.com/questions/300170/ideals-of-mathbbzx – Pythagoras Nov 12 '19 at 18:14
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Agha Farshad.. koja gir kardi?! – homosapien Nov 15 '23 at 22:17