Examples of 'artinian' in a sentence
Meaning of "artinian"
artinian (adjective) - In the context of mathematics or algebra, 'Artinian' is used to describe a type of ring or module that satisfies certain descending chain conditions. It is related to the mathematical concept of Artinian rings or modules
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- in which any descending chain of ideals eventually starts repeating.
- in which any descending chain of submodules eventually starts repeating.
How to use "artinian" in a sentence
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artinian
A ring is semisimple if and only if it is artinian and is semiprimitive.
Artinian local ring.
M has a composition series if and only if it is noetherian and artinian.
An algebra over a field k is artinian if and only if it has finite dimension.
Then it is also not left Artinian.
An Artinian ring is a ring satisfying the descending chain condition on ideals.
The analogous statement for left Artinian rings holds as well.
An Artinian ring is initially understood via its largest semisimple quotient.
A ring that is a semisimple module over itself is known as an Artinian semisimple ring.
An Artinian local ring is complete.
Composition series may thus be used to define invariants of finite groups and Artinian modules.
Also a simple Artinian ring.
In a right artinian ring, any nil ideal is nilpotent.
A quotient and localization of an Artinian ring is Artinian.
R is called right Artinian when this right module R is an Artinian module.
See also
Semisimple rings are necessarily Artinian rings.
Left Artinian ring.
Let M be a left module over a left Artinian ring.
Simple Artinian ring.
Lemma 5 can not be generalized to arbitrary artinian rings.
Conversely, every Artinian integral domain is a field.
A " is a finite product of commutative Artinian local rings.
In an Artinian ring, each element is either invertible or a zero divisor.
Modules of finite uniform dimension generalize both Artinian modules and Noetherian modules.
If an Artinian semisimple ring contains a field as a central subring, it is called a semisimple algebra.
Semisimple rings are both Artinian and Noetherian.
Despite the two conditions appearing symmetric, Noetherian rings are much more general than Artinian rings.
Notice that this generalizes the classification of simple artinian rings given by Artin-Wedderburn theory.
A ring is Artinian if and only if it is Noetherian and its Krull dimension is ≤ 0.
Her research area is representation theory for Artinian algebras, commutative algebra, and homological algebra.
The ring of integers formula 6 is a Noetherian ring but is not Artinian.
It is a somewhat surprising fact that a left Artinian ring is left Noetherian the Hopkins-Levitzki theorem.
A ring with finitely many, say left, ideals is left Artinian.
Notice that this generalizes the classification of simple artinian rings given by Artin - Wedderburn theory.
The term "Weil algebra" is also sometimes used to mean a finite-dimensional real local Artinian ring.
The Artin-Wedderburn theorem characterizes all simple Artinian rings as the matrix rings over a division ring.
M is Artinian if and only if every quotient module M / N is f . cog.
In particular, commutative Artinian rings are right and left Kasch.
Notably, the Artin-Wedderburn theorem 's conclusion about the structure of simple right Artinian rings is recovered.
The episode was directed by Liz Artinian ( art director ) and Joe Johnston supervising director.
As a consequence, any finitely-generated module over an Artinian ring is Artinian.
Then formula 2 is Artinian for every positive integer " n.
A consequence of the Akizuki-Hopkins-Levitzki Theorem is that every left Artinian ring is left Noetherian.
The zero ring is Artinian and ( therefore ) Noetherian.
Semi-hereditary Orders in a, simple Artinian ring.
The definition of " left Artinian ring " is done analogously.
If " I " is a nonzero ideal of a Dedekind domain " A ", then formula 3 is a principal Artinian ring.
AT this point, Mr. Artinian was prepared to believe the worst.
In fact, by the Hopkins - Levitzki theorem, every Artinian ring is Noetherian.
Modules over Artinian rings = = Let " M " be a left module over a left Artinian ring.
