Artinian local ring

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

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.