In
mathematics, specifically
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
, an Artinian ring (sometimes Artin ring) is a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
that satisfies the
descending chain 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 con ...
on (one-sided)
ideals; 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 ring
In mathematics, more specifically abstract algebra, a finite ring is a ring that has a finite number of elements.
Every finite field is an example of a finite ring, and the additive part of every finite ring is an example of an abelian finite grou ...
s and rings that are
finite-dimensional vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s over
fields
Fields may refer to:
Music
* Fields (band), an indie rock band formed in 2006
* Fields (progressive rock band), a progressive rock band formed in 1971
* ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010)
* "Fields", a song b ...
. The definition of Artinian rings may be restated by interchanging the descending chain condition with an equivalent notion: the
minimum condition.
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
Simple or SIMPLE may refer to:
*Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by Johnn ...
Artinian ring as a
ring of matrices over a
division ring
In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element ...
. This implies that a simple ring is left Artinian
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is b ...
it is right Artinian.
The same definition and terminology can be applied to
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Mo ...
s, with ideals replaced by
submodule
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the mo ...
s.
Although the descending chain condition appears dual to the
ascending chain 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 con ...
, 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
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noethe ...
. This is not true for general modules; that is, an
Artinian module
In mathematics, specifically 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 ...
need not be a
Noetherian module In abstract algebra, a Noetherian module is a module that satisfies the ascending chain condition on its submodules, where the submodules are partially ordered by inclusion.
Historically, Hilbert was the first mathematician to work with the proper ...
.
Examples and counterexamples
*An
integral domain
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
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
In mathematics, more specifically abstract algebra, a finite ring is a ring that has a finite number of elements.
Every finite field is an example of a finite ring, and the additive part of every finite ring is an example of an abelian finite grou ...
(e.g.,
) is left and right Artinian.
*Let ''k'' be a field. Then
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 languag ...
''n''.
*Similarly,
is an Artinian ring with
maximal ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals c ...
.
*Let
be an endomorphism between a finite-dimensional vector space ''V''. Then the subalgebra
generated by
is a commutative Artinian ring.
*If ''I'' is a
nonzero ideal of a
Dedekind domain
In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily ...
''A'', then
is a
principal Artinian ring.
*For each
, the full
matrix ring 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
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...
''R''
'x''is not Artinian, since the ideal generated by
is (properly) contained in the ideal generated by
for all
natural numbers ''n''. In contrast, if ''R'' is Noetherian so is ''R''
'x''by the
Hilbert basis theorem.
*The ring of integers
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 (i.e., has
composition series In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many natura ...
), (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
Product may refer to:
Business
* Product (business), an item that serves as a solution to a specific consumer problem.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
Mathematics
* Produ ...
of commutative Artinian
local rings.
*''A'' / nil(''A'') is a
semisimple ring
In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A ring that is a semisimple module over itsel ...
, where nil(''A'') is the
nilradical of ''A''.
* Every finitely generated module over ''A'' has finite length. (see above)
*''A'' has
Krull dimension
In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally th ...
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 definition y ...
since
prime ideals are maximal.)
*
is finite and discrete.
*
is discrete.
Let ''k'' be a field and ''A'' finitely generated ''k''-
algebra
Algebra () is one of the 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 mathematics.
Elementary ...
. Then ''A'' is Artinian if and only if ''A'' is finitely generated as ''k''-module.
An Artinian local ring is complete. A
quotient
In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
and
localization of an Artinian ring is Artinian.
Simple Artinian ring
A
simple
Simple or SIMPLE may refer to:
*Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by Johnn ...
Artinian ring ''A'' is a matrix ring over a division ring. Indeed,
let ''I'' be a minimal (nonzero) right ideal of ''A''. Then, since
is a two-sided ideal,
since ''A'' is simple. Thus, we can choose
so that
. Assume ''k'' is minimal with respect that property. Consider the map of right ''A''-modules:
:
It is
surjective. If it is not
injective, then, say,
with nonzero
. Then, by the minimality of ''I'', we have:
. It follows:
:
,
which contradicts the minimality of ''k''. Hence,
and thus
.
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.
Every Artin algebra is an Artin ring.
Dual and transpose
There are several different d ...
*
Artinian ideal In abstract algebra, an Artinian ideal, named after Emil Artin, is encountered in ring theory, in particular, with polynomial rings.
Given a polynomial ring ''R'' = ''k'' 'X''1, ... ''X'n''where ''k'' is some field, an Arti ...
*
Serial module
In abstract algebra, a uniserial module ''M'' is a module over a ring ''R'', whose submodules are totally ordered by inclusion. This means simply that for any two submodules ''N''1 and ''N''2 of ''M'', either N_1\subseteq N_2 or N_2\subseteq N_1. ...
*
Semiperfect ring
*
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 is ...
*
Noetherian ring
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noethe ...
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