Example of a ring such that it is local with nilpotent nilradical but not Artinian

Question

If $(R,m)$ be an Artinian ring then we know that $m^n=0$ for some integer $n$. Now if $(R,m)$ be a ring such that $m^n=0$, is this Artinian? If no, please give me an example. thanks

Answer 1

No. Take $R=k[X_1,X_2,\dots]/(X_1,X_2,\dots)^2$. It is not noetherian (so not artinian), but it is local with nilpotent maximal ideal $m=(x_1,x_2,\dots)$ (here $x_i$ denotes the equivalence class of $X_i$ in $R$).