- State the definition of a Noetherian ring (give all three equivalent characterizations).
- Give standard examples and non-examples of Noetherian rings.
Solution
For a ring \(R\), the following are equivalent:
- every ascending chain of ideals stabilizes;
- every ideal of \(R\) is finitely generated;
- every nonempty set of ideals has a maximal element under inclusion.
Indeed, (1)\(\Rightarrow\)(2): if an ideal \(I\) were not finitely generated, choose successively \(x_i\in I\) outside \((x_1,\dots,x_{i-1})\) to obtain a strictly ascending chain. For (2)\(\Rightarrow\)(1), if \(I_1\subseteq I_2\subseteq\cdots\), then \(I=\bigcup I_n\) is an ideal; finite generation of \(I\) places all generators in some \(I_N\), so \(I=I_N\) and the chain stabilizes. The equivalence with (3) is the standard maximal-condition reformulation of ACC.
Examples: fields and PIDs are Noetherian; Hilbert’s basis theorem shows that \(R[x_1,\dots,x_n]\) is Noetherian whenever \(R\) is; quotients and localizations of Noetherian rings are Noetherian. In particular \(k[[x_1,\dots,x_n]]\) is Noetherian.
A standard nonexample is \(k[x_1,x_2,\dots]\), since \[ (x_1)\subsetneq(x_1,x_2)\subsetneq\cdots \] does not stabilize.