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In
abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...
, an Artinian module is a
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 * Modula ...
that satisfies the
descending chain conditionIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
on its poset of submodules. They are for modules what
Artinian ring In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), r ...
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 (; March 3, 1898 – December 20, 1962) was an Austrian Austrian may refer to: * Austrians, someone from Austria or of Austrian descent ** Someone who is considered an Austrian citizen, see Austrian nationality law * Something associa ...
. In the presence of the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

axiom of choice
, the descending chain condition becomes equivalent to the
minimum conditionIn mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly Ideal (ring theory), ideals in certain commutative rings.Jacobson (2009), p. 1 ...
, and so that may be used in the definition instead. Like
Noetherian modulesIn abstract algebra, a Noetherian module is a module (mathematics), module that satisfies the ascending chain condition on its submodules, where the submodules are partially ordered by inclusion (set theory), inclusion. Historically, David Hilbert, ...
, 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 In mathematics, a finitely generated module is a module (mathematics), module that has a Finite set, finite generating set. A finitely generated module over a Ring (mathematics), ring ''R'' may also be called a finite ''R''-module, finite over ''R'' ...
over an Artinian ring is Artinian.Lam (2001), Proposition 1.21, p. 19 Since an Artinian ring is also a Noetherian ring, 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 length of a module, finite length; however, if ''R'' is not Artinian, or if ''M'' is not finitely generated, there are #Relation to the Noetherian condition, 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 Artinian ring, Artinian when this right module ''R'' is an Artinian module. The definition of "left Artinian ring" is done analogously. 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 as 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 ideal (ring theory), 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.


Relation to the Noetherian condition

Unlike the case of rings, there are Artinian modules which are not Noetherian modules. For example, consider the ''p''-primary component of \mathbb/\mathbb, that is \mathbb[1/p] / \mathbb, which is isomorphic to the ''p''-quasicyclic group \mathbb(p^\infty), regarded as \mathbb-module. The chain \langle 1/p \rangle \subset \langle 1/p^2 \rangle \subset \langle 1/p^3 \rangle \subset \cdots does not terminate, so \mathbb(p^\infty) (and therefore \mathbb/\mathbb) is not Noetherian. Yet every descending chain of (without loss of generality) proper submodules terminates: Each such chain has the form \langle 1/n_1 \rangle \supseteq \langle 1/n_2 \rangle \supseteq \langle 1/n_3 \rangle\supseteq \cdots for some integers n_1 , n_2 , n_3,\ldots , and the inclusion of \langle 1/n_ \rangle \subseteq \langle 1/n_i \rangle implies that n_ must divide n_i. So n_1 , n_2 , n_3, \ldots is a decreasing sequence of positive integers. Thus the sequence terminates, making \mathbb(p^\infty) Artinian. Over a commutative ring, every cyclic Artinian module is also Noetherian, but over noncommutative rings cyclic Artinian modules can have uncountable length of a module, length as shown in the article of Hartley and summarized nicely in the Paul Cohn article dedicated to Hartley's memory. Another relevant result is the Akizuki–Hopkins–Levitzki theorem, which states that the Artinian and Noetherian conditions are equivalent for modules over a semiprimary ring.


See also

* Noetherian module * ascending chain condition, Ascending/Descending chain condition * Composition series * Krull dimension


Notes


References

* * * * {{cite book , last = Lam , first = T.Y. , title = A First Course in Noncommutative Rings , isbn = 978-0-387-95325-0 , year = 2001 , publisher = Springer Verlag , chapter = Chapter 1. Wedderburn-Artin theory Module theory