Artinian Ring
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, specifically
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The te ...
, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided)
ideals Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considered ...
; that is, there is no infinite descending sequence of ideals. Artinian rings are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are finite-dimensional vector spaces over fields. The definition of Artinian rings may be restated by interchanging the descending chain condition with an equivalent notion: the
minimum condition In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly ideals in certain commutative rings.Jacobson (2009), p. 142 and 147 These c ...
. Precisely, 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 the left and right definitions coincide, but in general they are distinct from each other. The Artin–Wedderburn theorem characterizes every simple Artinian ring as a ring of matrices over a division ring. This implies that a simple ring is left Artinian if and only if it is right Artinian. The same definition and terminology can be applied to modules, with ideals replaced by submodules. Although the descending chain condition appears dual to the ascending chain condition, in rings it is in fact the stronger condition. Specifically, a consequence of the Akizuki–Hopkins–Levitzki theorem is that a left (resp. right) Artinian ring is automatically a left (resp. right) Noetherian ring. This is not true for general modules; that is, an Artinian module need not be a Noetherian module.


Examples and counterexamples

*An integral domain is Artinian if and only if it is a field. *A ring with finitely many, say left, ideals is left Artinian. In particular, a finite ring (e.g., \mathbb/n \mathbb) is left and right Artinian. *Let ''k'' be a field. Then k (t^n) is Artinian for every positive
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
''n''. *Similarly, k ,y(x^2, y^3, xy^2) = k \oplus k\cdot x \oplus k \cdot y \oplus k\cdot xy \oplus k \cdot y^2 is an Artinian ring with maximal ideal (x,y). *Let x be an endomorphism between a finite-dimensional vector space ''V''. Then the subalgebra A \subset \operatorname(V) generated by x is a commutative Artinian ring. *If ''I'' is a nonzero ideal of a Dedekind domain ''A'', then A/I is a principal Artinian ring. *For each n \ge 1, the full
matrix ring In abstract algebra, a matrix ring is a set of matrices with entries in a ring ''R'' that form a ring under matrix addition and matrix multiplication . The set of all matrices with entries in ''R'' is a matrix ring denoted M''n''(''R'')Lang, ''U ...
M_n(R) over a left Artinian (resp. left Noetherian) ring ''R'' is left Artinian (resp. left Noetherian). The following two are examples of non-Artinian rings. *If ''R'' is any ring, then the polynomial ring ''R'' 'x''is not Artinian, since the ideal generated by x^ is (properly) contained in the ideal generated by x^n for all
natural numbers In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal n ...
''n''. In contrast, if ''R'' is Noetherian so is ''R'' 'x''by the Hilbert basis theorem. *The ring of integers \mathbb is a Noetherian ring but is not Artinian.


Modules over Artinian rings

Let ''M'' be a left module over a left Artinian ring. Then the following are equivalent ( Hopkins' theorem): (i) ''M'' is finitely generated, (ii) ''M'' has
finite length In abstract algebra, the length of a module is a generalization of the dimension of a vector space which measures its size. page 153 In particular, as in the case of vector spaces, the only modules of finite length are finitely generated modules. It ...
(i.e., has composition series), (iii) ''M'' is Noetherian, (iv) ''M'' is Artinian.


Commutative Artinian rings

Let ''A'' be a commutative Noetherian ring with unity. Then the following are equivalent. *''A'' is Artinian. *''A'' is a finite product of commutative Artinian
local rings In abstract algebra, more specifically ring theory, local rings are certain ring (mathematics), rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on algebraic variety, vari ...
. *''A'' / nil(''A'') is a semisimple ring, where nil(''A'') is the nilradical of ''A''. * Every finitely generated module over ''A'' has finite length. (see above) *''A'' has Krull dimension zero. (In particular, the nilradical is the
Jacobson radical In mathematics, more specifically ring theory, the Jacobson radical of a ring R is the ideal consisting of those elements in R that annihilate all simple right R- modules. It happens that substituting "left" in place of "right" in the definitio ...
since
prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together ...
s are maximal.) *\operatornameA is finite and discrete. *\operatornameA is discrete. Let ''k'' be a field and ''A'' finitely generated ''k''-
algebra Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...
. Then ''A'' is Artinian if and only if ''A'' is finitely generated as ''k''-module. An Artinian local ring is complete. A quotient and localization of an Artinian ring is Artinian.


Simple Artinian ring

A simple Artinian ring ''A'' is a matrix ring over a division ring. Indeed, let ''I'' be a minimal (nonzero) right ideal of ''A''. Then, since AI is a two-sided ideal, AI = A since ''A'' is simple. Thus, we can choose a_i \in A so that 1 \in a_1 I + \cdots + a_k I. Assume ''k'' is minimal with respect that property. Consider the map of right ''A''-modules: :\begin I^ \to A, \\ (y_1, \dots, y_k) \mapsto a_1y_1 + \cdots + a_k y_k \end It is
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
. If it is not
injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
, then, say, a_1y_1 = a_2y_2 + \cdots + a_k y_k with nonzero y_1. Then, by the minimality of ''I'', we have: y_1 A = I. It follows: :a_1 I = a_1 y_1 A \subset a_2 I + \cdots + a_k I, which contradicts the minimality of ''k''. Hence, I^ \simeq A and thus A \simeq \operatorname_A(A) \simeq M_k(\operatorname_A(I)).


See also

*
Artin algebra In algebra, an Artin algebra is an algebra Λ over a commutative Artin ring ''R'' that is a finitely generated ''R''-module. They are named after Emil Artin Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician of Arm ...
* Artinian ideal * Serial module *
Semiperfect ring In the area of abstract algebra known as ring theory, a left perfect ring is a type of ring in which all left modules have projective covers. The right case is defined by analogy, and the condition is not left-right symmetric; that is, there ...
*
Gorenstein ring In commutative algebra, a Gorenstein local ring is a commutative Noetherian local ring ''R'' with finite injective dimension as an ''R''-module. There are many equivalent conditions, some of them listed below, often saying that a Gorenstein ring ...
* Noetherian ring


Notes


References

* * * Charles Hopkins. Rings with minimal condition for left ideals. Ann. of Math. (2) 40, (1939). 712–730. * * * {{cite book , last1=Brešar, first1=Matej, title=Introduction to Noncommutative Algebra , year=2014 , publisher=Springer , isbn=978-3-319-08692-7 Ring theory