Artin rings

Artin rings are a class of algebraic structures used in advanced mathematics, particularly in algebraic geometry and commutative algebra. They are rings (systems where addition and multiplication are defined) that have a finite-dimensional structure over a field, with a special property that their maximal ideals (a type of subset) are nilpotent—meaning some power of the ideal equals zero. These rings often model "local" behaviors of geometric objects at specific points, capturing small-scale, infinitesimal information. They are essential tools for understanding how complex shapes behave near particular locations.