Noetherian rings

There are two equivalent definitions of Noetherian ring: A commutative ring with unity is called Noetherian if

1. every ideal of $A$ is finitely generated.
2. every ascending chain of ideals in $A$ stabilizes: that is, for every increasing sequence $I_1\subseteq I_2\subseteq \dots$ of ideals in $A$, there exists an $n$ such that $I_n=I_{n+1}=\dots$.

$2⟹1$: Suppose $I$ is not finitely generated. For any finite set $\{a_1,\dots ,a_n\}$, $⟨a_1,\dots ,a_n⟩\ne I$. Consider the chain

$$⟨a_1⟩\subset ⟨a_1,a_2⟩\subset \dots .$$

$1⟹2$: Let there be a chain of ideals $I_1\subset I_2\subset \dots$. Then $J={\bigcup }_{n\ge 1}{I}_n$ is an ideal. Suppose $J$ is finitely generated by $\{b_1,\dots ,b_m\}$. $b_i\in I_j$ for some $j$ for each $i$. It follows that $\{b_1,\dots ,b_m\}\in I_k$ for some $I_k$. It follows that $I_k=(b_1,\dots ,b_m)$, and for all $l\ge k$, we have $I_l=I_k$.

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Theorem 256.1(Hilbert's basis theorem).

If $R$ is noetherian then $R[x]$ is noetherian.

Thus, $\mathbb{Z}[x_1,\dots ,x_n]$ is noetherian. $k[x_1,\dots ,x_n]$ is noetherian for any field $k$. If $R$ is noetherian, for any ideal $I$ of $R$, $R/I$ is noetherian.

Also, $R=k[x,y]$ is noetherian. However, the subring $S=k[x,xy,x{y}^2,\dots ]$ is not noetherian: $⟨x⟩\subset ⟨x,xy⟩\subset ⟨x,x{y}^2⟩\subset \dots$ .