Is there a local ring but not Noetherian?

Question

I was looking for an example of a non-Noetherian local commutative ring with $1$.

I would appreciate if anyone can point to a reference.

Answer 1

Take a field $k$ and the ring $k[X_1,\ldots ,X_n,\ldots ]$. Localize it at the maximal ideal $(X_1,\ldots , X_n,\ldots )$. Then the ideals $(X_1)\subset (X_1,X_2)\subset ... \subset (X_1,\ldots , X_n) \subset \ldots$ still form a strictly increasing union of ideals, and so the localization is local but not noetherian

(To see that the inclusions are strict, for instance kill everyone but $X_n$ and then include in power series to get a map $k[(X_i)] \to k[[X_n]]$ that sends anyone $\notin$ the maximal ideal (that is, anyone with a constant term) to an invertible element, and so that factors through our ring, and sends $(X_1,\ldots , X_{n-1})$ to $0$ but not $X_n$).