ARTINIAN Meaning and
Definition
-
Artinian is an adjective used to describe a mathematical ring that satisfies a certain property. In abstract algebra, a ring is a set equipped with two binary operations, usually addition and multiplication, which must satisfy certain axioms. A ring is said to be Artinian if it has a property called the Ascending Chain Condition (ACC) on its ideals.
The ACC on ideals means that in an Artinian ring, it is impossible to have an infinite increasing chain of ideals. More precisely, for every sequence I₁ ⊂ I₂ ⊂ I₃ ⊂ ⋯ of ideals in the Artinian ring, there exists some positive integer n such that Iₙ = Iₙ₊₁ = Iₙ₊₂ = ⋯. In simpler terms, this condition ensures that there will always be a point, after a certain number of steps, where the ideals stop getting strictly bigger.
This property is named after the mathematician Emil Artin who made important contributions to abstract algebra. Artinian rings have many interesting properties and are widely studied in algebraic geometry and other areas of algebra. They provide a rich framework for understanding various mathematical structures such as commutative rings, noncommutative rings, and modules. The concept of Artinian rings generalizes the notion of finite-dimensional vector spaces in linear algebra, making it a fundamental notion in algebraic theory and its applications.
