Artinian ring

# Artinian ring

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In abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...

, an Artinian ring is a ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...

that satisfies the descending chain condition on ideals
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....

. They are also called Artin rings and are named after Emil Artin
Emil Artin
Emil Artin was an Austrian-American mathematician of Armenian descent.-Parents:Emil Artin was born in Vienna to parents Emma Maria, née Laura , a soubrette on the operetta stages of Austria and Germany, and Emil Hadochadus Maria Artin, Austrian-born of Armenian descent...

, who first discovered that the descending chain condition for ideals simultaneously generalizes finite ring
Finite ring
In mathematics, more specifically abstract algebra, a finite ring is a ring that has a finite number of elements....

s and rings that are finite-dimensional vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

s over fields
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

.

A ring is left Artinian if it satisfies the descending chain condition on left ideals, right Artinian if it satisfies the descending chain condition on right ideals, and Artinian or two-sided Artinian if it is both left and right Artinian. For commutative rings and for the two classes of rings mentioned above these concepts coincide, but in general they are different.

The Artin–Wedderburn theorem
Artin–Wedderburn theorem
In abstract algebra, the Artin–Wedderburn theorem is a classification theorem for semisimple rings. The theorem states that an Artinian semisimple ring R is isomorphic to a product of finitely many ni-by-ni matrix rings over division rings Di, for some integers ni, both of which are uniquely...

characterizes all simple
Simple ring
In abstract algebra, a simple ring is a non-zero ring that has no ideal besides the zero ideal and itself. A simple ring can always be considered as a simple algebra. This notion must not be confused with the related one of a ring being simple as a left module over itself...

artinian rings as the matrix ring
Matrix ring
In abstract algebra, a matrix ring is any collection of matrices forming a ring under matrix addition and matrix multiplication. The set of n×n matrices with entries from another ring is a matrix ring, as well as some subsets of infinite matrices which form infinite matrix rings...

s over a division ring
Division ring
In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a non-trivial ring in which every non-zero element a has a multiplicative inverse, i.e., an element x with...

. This implies that for simple rings, both left and right Artinian coincide.

Although the descending chain condition appears similar to the ascending chain condition
Ascending chain condition
The ascending chain condition and descending chain condition are finiteness properties satisfied by some algebraic structures, most importantly, ideals in certain commutative rings...

, in rings it is in fact the stronger condition. Specifically, a consequence of the Akizuki–Hopkins–Levitzki theorem
Hopkins–Levitzki theorem
In the branch of abstract algebra called ring theory, the Akizuki-Hopkins–Levitzki theorem connects the descending chain condition and ascending chain condition in modules over semiprimary rings. A ring R is called semiprimary if R/J is semisimple and J is a nilpotent ideal, where J denotes the...

is that a left (right) Artinian ring is automatically a left (right) Noetherian ring
Noetherian ring
In mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, is a ring in which every non-empty set of ideals has a maximal element...

. This is not true for general modules, that is, an Artinian module
Artinian module
In abstract algebra, an Artinian module is a module that satisfies the descending chain condition on its poset of submodules. They are for modules what Artinian rings are for rings, and a ring is Artinian if and only if it is an Artinian module over itself...

need not be a Noetherian module
Noetherian module
In abstract algebra, an Noetherian module is a module that satisfies the ascending chain condition on its submodules, where the submodules are partially ordered by inclusion....

.

A direct characterization of commutative
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....

Artinian rings is that R is Artinian if and only if it is Noetherian and if R / nil(R) is isomorphic to a direct product
Direct product
In mathematics, one can often define a direct product of objectsalready known, giving a new one. This is generally the Cartesian product of the underlying sets, together with a suitably defined structure on the product set....

of finitely many fields, where nil(R) is the nilradical of R.