Artin ring

An Artin ring is a type of mathematical structure used in algebra to study solutions to equations and their properties. Specifically, it’s a finite-dimensional ring over a field that has a well-behaved, "finite" structure. These rings include examples like the ring of formal power series with a finite number of variables. Artin rings are important because they help mathematicians understand local properties of algebraic objects, such as how shapes or solutions behave near specific points, by providing a manageable context for complex algebraic and geometric problems.