It is true that for a polynomial ring $R[x]$, there will be a prime element?
2026-04-01 03:39:35.1775014775
Any polynomial ring contains a prime element?
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The answer is still no (even though one comment suggested otherwise), consider $R=\mathbb C[x,y]/(x^2,xy,y^2)$ and $R[t] = \mathbb C[x,y,t]/(x^2,xy,y^2)$
All prime ideals of $R[t]$ are of the form $(x,y,t-c)$ or $(x,y)$.