Let $R[x_1,\ldots, x_n]$ be a polynomial ring over an integral domain $R$. When is the ideal $(r_1x_1+\ldots+r_nx_n)$ for some $r_1,\ldots,r_n \in R$ prime?
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If $D$ is a domain then $, f,$ is prime in $,D[x] \iff!\underbrace{f,\ \rm is\ prime\ in\ K[x]}_{\large\iff f\rm\ is\ irreducible\quad }!$ and $,f,$ is superprimitive. $\ \ $ – Bill Dubuque Jan 28 '23 at 16:06
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Thanks. We can apply induction by number of variables. But, is there a condition in terms of $r_1,\ldots,r_n$? – Constantine Jan 28 '23 at 17:02