Say we have $R$ commutative with 1. Taking $R[x]$ as the ring of polynomials, how would you show that an ideal $I$ for $R$ is prime $\iff$ an ideal $I[x] = \{\sum\limits_{i=0}^na_ix^i | a_i \in I\}$ of $R[x]$ is prime?
2026-03-25 23:10:41.1774480241
Prime ideals in ring and polynomial rings
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Hint : $R[x]/I[x] \simeq (R/I)[x]$. How can you characterize the primeness of an ideal with the quotient by this ideal ?