Artinian module
Encyclopedia
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 ring
s are for rings, and a ring is Artinian if and only if it is an Artinian module over itself (with left or right multiplication). Both concepts are named for Emil Artin
.
Like Noetherian modules, Artinian modules enjoy the following heredity property:
The converse also holds:
As a consequence, any finitely-generated module over an Artinian ring is Artinian. Since an Artinian ring is also Noetherian, and finitely-generated modules over a Noetherian ring are Noetherian, it is true that for an Artinian ring R, any finitely-generated R-module is both Noetherian and Artinian, and is said to be of finite length
; however, if R is not Artinian, or if M is not finitely generated, there are counterexamples.
when this right module R is an Artinian module. The definition of "left Artinian ring" is done analagously. For noncommutative rings this distinction is necessary, because it is possible for a ring to be Artinian on one side only.
The left-right adjectives are not normally necessary for modules, because the module M is usually given as a left or right R module at the outset. However, it is possible that M may have both a left and right R module structure, and then calling M Artinian is ambiguous, and it becomes necessary to clarify which module structure is Artinian. To separate the properties of the two structures, one can abuse terminology and refer to M as left Artinian or right Artinian when, strictly speaking, it is correct to say that M, with its left R-module structure, is Artinian.
The occurrence of modules with a left and right structure is not unusual: for example R itself has a left and right R module structure. In fact this is an example of a bimodule
, and it may be possible for an abelian group M to be made into a left-R, right-S bimodule for a different ring S. Indeed, for any right module M, it is automatically a left module over the ring of integers Z, and moreover is a Z-R bimodule. For example, consider the rational numbers Q as a Z-Q bimodule in the natural way. Then Q is not Artinian as a left Z module, but it is Artinian a right Q module.
The Artinian condition can be defined on bimodule structures as well: an Artinian bimodule is a bimodule
whose poset of sub-bimodules satisfies the descending chain condition. Since a sub-bimodule of an R-S bimodule M is a fortiori a left R-module, if M considered as a left R module were Artinian, then M is automatically an Artinian bimodule. It may happen, however, that a bimodule is Artinian without its left or right structures being Artinian, as the following example will show.
Example: It is well known that a simple ring
is left Artinian if and only if it is right Artinian, in which case it is a semisimple ring. Let R be a simple ring which is not right Artinian. Then it is also not left Artinian. Considering R as an R-R bimodule in the natural way, its sub-bimodules are exactly the ideals
of R. Since R is simple there are only two: R and the zero ideal. Thus the bimodule R is Artinian as a bimodule, but not Artinian as a left or right R-module over itself.
s. For example, consider the p-primary component of
, that is 
, which is isomorphic to the p-quasicyclic group 
, regarded as 
-module. The chain 
does not terminate, so 
(and therefore 
) is not Noetherian. Yet every descending chain of (without loss of generality) proper submodules terminates: Each such chain has the form 
for some integers 
..., and the inclusion of 
implies that 
must divide 
. So 
... is a decreasing sequence of positive integers. Thus the sequence terminates, making 
Artinian.
Over a commutative ring, every cyclic Artinian module is also Noetherian, but over noncommutative rings cyclic Artinian modules can have uncountable length
as shown in the article of Hartley and summarized nicely in the Paul Cohn
article dedicated to Hartley's memory.
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 module is a module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
that satisfies the descending chain condition on its poset of submodules. They are for modules what Artinian ring
Artinian ring
In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. They are also called Artin rings and are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are...
s are for rings, and a ring is Artinian if and only if it is an Artinian module over itself (with left or right multiplication). Both concepts are named for 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...
.
Like Noetherian modules, Artinian modules enjoy the following heredity property:
- If M is an Artinian R-module, then so is any submodule and any quotient of M.
The converse also holds:
- If M is any R module and N any Artinian submodule such that M/N is Artinian, then M is Artinian.
As a consequence, any finitely-generated module over an Artinian ring is Artinian. Since an Artinian ring is also Noetherian, and finitely-generated modules over a Noetherian ring are Noetherian, it is true that for an Artinian ring R, any finitely-generated R-module is both Noetherian and Artinian, and is said to be of finite length
Length of a module
In abstract algebra, the length of a module is a measure of the module's "size". It is defined to be the length of the longest chain of submodules and is a generalization of the concept of dimension for vector spaces...
; however, if R is not Artinian, or if M is not finitely generated, there are counterexamples.
Left and right Artinian rings, modules and bimodules
The ring R can be considered as a right module, where the action is the natural one given by the ring multiplication on the right. R is called right ArtinianArtinian ring
In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. They are also called Artin rings and are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are...
when this right module R is an Artinian module. The definition of "left Artinian ring" is done analagously. For noncommutative rings this distinction is necessary, because it is possible for a ring to be Artinian on one side only.
The left-right adjectives are not normally necessary for modules, because the module M is usually given as a left or right R module at the outset. However, it is possible that M may have both a left and right R module structure, and then calling M Artinian is ambiguous, and it becomes necessary to clarify which module structure is Artinian. To separate the properties of the two structures, one can abuse terminology and refer to M as left Artinian or right Artinian when, strictly speaking, it is correct to say that M, with its left R-module structure, is Artinian.
The occurrence of modules with a left and right structure is not unusual: for example R itself has a left and right R module structure. In fact this is an example of a bimodule
Bimodule
In abstract algebra a bimodule is an abelian group that is both a left and a right module, such that the left and right multiplications are compatible...
, and it may be possible for an abelian group M to be made into a left-R, right-S bimodule for a different ring S. Indeed, for any right module M, it is automatically a left module over the ring of integers Z, and moreover is a Z-R bimodule. For example, consider the rational numbers Q as a Z-Q bimodule in the natural way. Then Q is not Artinian as a left Z module, but it is Artinian a right Q module.
The Artinian condition can be defined on bimodule structures as well: an Artinian bimodule is a bimodule
Bimodule
In abstract algebra a bimodule is an abelian group that is both a left and a right module, such that the left and right multiplications are compatible...
whose poset of sub-bimodules satisfies the descending chain condition. Since a sub-bimodule of an R-S bimodule M is a fortiori a left R-module, if M considered as a left R module were Artinian, then M is automatically an Artinian bimodule. It may happen, however, that a bimodule is Artinian without its left or right structures being Artinian, as the following example will show.
Example: It is well known that a simple ring
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...
is left Artinian if and only if it is right Artinian, in which case it is a semisimple ring. Let R be a simple ring which is not right Artinian. Then it is also not left Artinian. Considering R as an R-R bimodule in the natural way, its sub-bimodules are exactly the 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"....
of R. Since R is simple there are only two: R and the zero ideal. Thus the bimodule R is Artinian as a bimodule, but not Artinian as a left or right R-module over itself.
Relation to the Noetherian condition
Unlike the case of rings, there are Artinian modules which are not Noetherian moduleNoetherian 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....
s. For example, consider the p-primary component of
, that is , which is isomorphic to the p-quasicyclic group , regarded as -module. The chain does not terminate, so (and therefore ) is not Noetherian. Yet every descending chain of (without loss of generality) proper submodules terminates: Each such chain has the form for some integers ..., and the inclusion of implies that must divide . So ... is a decreasing sequence of positive integers. Thus the sequence terminates, making Artinian.Over a commutative ring, every cyclic Artinian module is also Noetherian, but over noncommutative rings cyclic Artinian modules can have uncountable length
Length of a module
In abstract algebra, the length of a module is a measure of the module's "size". It is defined to be the length of the longest chain of submodules and is a generalization of the concept of dimension for vector spaces...
as shown in the article of Hartley and summarized nicely in the Paul Cohn
Paul Cohn
Paul Moritz Cohn FRS was Astor Professor of Mathematics at University College London, 1986-9, and author of many textbooks on algebra...
article dedicated to Hartley's memory.
See also
- Noetherian moduleNoetherian moduleIn 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....
- Ascending/Descending chain conditionAscending chain conditionThe ascending chain condition and descending chain condition are finiteness properties satisfied by some algebraic structures, most importantly, ideals in certain commutative rings...
- Composition seriesComposition seriesIn 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 naturally occurring modules are not semisimple, hence...
- Krull dimensionKrull dimensionIn commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull , is the supremum of the number of strict inclusions in a chain of prime ideals. The Krull dimension need not be finite even for a Noetherian ring....