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prove the converse to Hilbert basis theoren: if the polynomial ring $R[x]$ is Noetherian, then $R$ is noetheian

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Any factor ring of a noetherian ring is noetherian. Since $x$ generates a two-sided ideal of $R[x]$ and $R[x]/(x)$ is isomorphic to $R$, then $R$ is noetherian. –  J. Gaddis Nov 19 '12 at 13:55
    
tank you ${{{{}}}}$ –  mshj Nov 19 '12 at 17:24
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1 Answer

Let $I\subseteq R$ an ideal. Then $J := I + X\cdot R[X]\subseteq R[X]$ is an ideal, hence finally generated, say $J = \langle p_0, \ldots, p_n\rangle$. Now let $a_i = p_i(0) \in R$ for $1 \le i \le n$. We have $\langle a_0, \ldots, a_n\rangle\subseteq I$ by definition of $J$, now let $a \in I$, then for some $f_i \in R[X]$ we have $$ a = \sum_{i=0}^n f_i p_i $$ which, evaluated at 0 gives $$ a = \sum_{i=0}^n f_i(0)a_i \in \langle a_0, \ldots, a_n \rangle $$ Hence $ I = \langle a_0,\ldots, a_n\rangle$ and $I$ is finitely generated.

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