In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ... In mathematics, a poset P is said to satisfy the ascending chain condition (ACC) if every ascending chain a1 ≤ a2 ≤ ... of elements of P is eventually stationary, that is, there is some positive integer n such that am = an for all m > n. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ...

There are two classes of rings that have very similar properties:

• Rings whose underlying set is finite.
• Rings that are finite-dimensional vector spaces over fields.

Emil Artin first discovered that the descending chain condition for ideals generalizes both classes of rings simultaneously. Artinian rings are named after him. A forgetful functor is a type of functor in mathematics. ... In mathematics, a set is called finite if and only if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... Jump to: navigation, search A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... Emil Artin (March 3, 1898-December 20, 1962) was an Austrian mathematician born in Vienna who spent his career in Germany (mainly in Hamburg) until the Nazi threat when he emigrated to the USA in 1937 where he was at Indiana University 1938-1946, and Princeton University 1946-1958. ...

For noncommutative rings, we need to distinguish three very similar concepts:

• A ring is left Artinian if it satisfies the descending chain condition on left ideals.
• A ring is right Artinian if it satisfies the descending chain condition on right ideals.
• A ring is Artinian or two-sided Artinian if it is both left and right Artinian.

For commutative rings, these concepts all coincide. They also coincide for the two classes of rings mentioned above, but in general they are different. There are rings that are left Artinian and not right Artinian, and vice versa.

The Artin-Wedderburn theorem characterizes all simple rings that are Artinian: they are the matrix rings over a division ring. This implies that for simple rings, both left and right Artinian coincide. In abstract algebra, the Artin-Wedderburn theorem is a classification theorem for semisimple product of ni-by-ni matrix rings over division rings Di, for some integers ni, both of which are uniquely determined up to permutation of the index i. ... In abstract algebra, a simple ring is a non-zero ring that has no ideal besides the zero ideal and itself. ... In abstract algebra the matrix ring M(n,R) is set of all n-by-n matrices over an arbitrary ring R. This forms a ring under matrix addition and multiplication. ... In abstract algebra, a division ring, also called a skew field, is a ring with 0 ≠ 1 and such that every non-zero element a has a multiplicative inverse (i. ...

By the Hopkins-Levitzski theorem, a left (right) Artinian ring is automatically a left (right) Noetherian ring. In abstract algebra, a Noetherian ring is a ring that satisfies the ascending chain condition on ideals. ...