Artinian local ring

An Artinian local ring is an Artinian ring with a unique maximal ideal. Its maximal ideal is nilpotent.

Let $(R,\mathfrak{m})$ be an Artinian local ring. If ${\dim }_{R/\mathfrak{m}}\mathfrak{m}/{\mathfrak{m}}^2=1$, then every ideal of $R$ is principal. The Nakayama lemma gives $\mathfrak{m}=(x)$, and nilpotence gives a finite chainFor a nonzero ideal $J$, choose the largest $r$ for which $J\subseteq (x^r)$ and take $y\in J\setminus (x^{r+1})$. Writing $y=x^{r}u$, one has $u\notin \mathfrak{m}$, so $u$ is a unit and $J=(x^r)=(y)$.

An Artinian local ring is a finite set exactly when its residue field is finite. If ${\mathfrak{m}}^r=0$, each layer ${\mathfrak{m}}^i/{\mathfrak{m}}^{i+1}$ is a finite-dimensional vector space over that residue field.