Let \(\displaystyle{\left(A,+,\cdot\right)}\) be the ring of germs of analytic functions \(\displaystyle{f:\mathbb{R}\to \mathbb{R}}\)

at \(\displaystyle{0\in\mathbb{R}}\). Then \(\displaystyle{\left(A,+,\cdot\right)}\) is a \(\displaystyle{\rm{Noethrian}}\)

local ring with maximal ideal \(\displaystyle{m=\langle{x\rangle}}\).