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How do you use Noetherian in a sentence? See 10+ example sentences showing how this word appears in different contexts, plus the exact meaning.

Rare word

Noetherian in a sentence

Noetherian meaning

  1. Satisfying some (usually ascending) chain condition:
  2. In which any ascending chain of ideals eventually becomes constant.

Using Noetherian

  • The main meaning on this page is: Satisfying some (usually ascending) chain condition: | In which any ascending chain of ideals eventually becomes constant. | Satisfying some (usually ascending) chain condition:
  • In the example corpus, noetherian often appears in combinations such as: noetherian ring, is noetherian, of noetherian.

Context around Noetherian

  • Average sentence length in these examples: 17.1 words
  • Position in the sentence: 4 start, 11 middle, 5 end
  • Sentence types: 20 statements, 0 questions, 0 exclamations

Often before:

left 3× commutative 1× artinian 1× non 1×

Often after:

ring 8× integral 2× commutative 1× unless 1× rings 1× projective 1×

Sentence patterns with Noetherian

These short patterns come directly from the example sentences and show how the word is often embedded:

of a noetherian ring is 2× are left noetherian and not 1× are right noetherian but not 1× be a noetherian commutative ring 1× for commutative noetherian rings this 1× for non noetherian integral domains 1× form a noetherian ring which 1× in a noetherian ring every 1×

Sentence openings and endings

Common sentence openings:

a unique factorization 1× a valuation ring 1× another consequence is 1× applications let be 1× bourbaki 7 3 1× equivalent conditions for 1×

Common sentence endings:

a noetherian ring 2× a local ring 1× and vice versa 1× completions are not 1× has finite height 1× hopkins levitzki theorem 1×

Words around Noetherian

Just before it in the word index:

noether 7 noesis 12 noesen 9 noes 7 noer 2 noen 5

Just after it in the word index:

noetic 5 nof 3 nofa 2 noffke 3 nofio 3 nofollow 3

Similar in corpus frequency:

aaj 26 abn 26 aboriginals 26 absalom 26 absolutism 26 abstractly 26

Corpus analysis for Noetherian

  • In this selection, "noetherian" usually appears in the middle of the sentence. The average example has 17.1 words, and this corpus slice is mostly made up of statements.
  • Around the word, left, commutative, artinian, ring, integral and commutative stand out and add context to how "noetherian" is used.
  • Recognizable usage signals include of a noetherian ring is and are left noetherian and not. That gives this page its own corpus information beyond isolated example sentences.
  • By corpus frequency, "noetherian" sits close to words such as aaj, abn and aboriginals, which helps place it inside the broader word index.

Example types with noetherian

The same corpus examples are grouped by length and sentence type, making it easier to see the contexts in which the word appears:

Short example sentences

Applications Let be a Noetherian commutative ring. (7 words)

If is a Noetherian ring, then is a Noetherian ring. (10 words)

A unique factorization domain is not necessarily a noetherian ring. (10 words)

Longer example sentences

Equivalent conditions for a ring to be a UFD A Noetherian integral domain is a UFD if and only if every height 1 prime ideal is principal (a proof is given below). (32 words)

The notion of a Noetherian ring is of fundamental importance in both commutative and noncommutative ring theory, due to the role it plays in simplifying the ideal structure of a ring. (31 words)

Indeed, there are rings that are right Noetherian, but not left Noetherian, so that one must be careful in measuring the "size" of a ring this way. (27 words)

Example sentences (20)

If is a Noetherian ring, then is a Noetherian ring.

Indeed, there are rings that are right Noetherian, but not left Noetherian, so that one must be careful in measuring the "size" of a ring this way.

Stated differently, the image of any surjective ring homomorphism of a Noetherian ring is Noetherian.

There are rings that are left-Noetherian and not right-Noetherian, and vice versa.

Another consequence is that a left Artinian ring is right Noetherian if and only if right Artinian.

Applications Let be a Noetherian commutative ring.

A unique factorization domain is not necessarily a noetherian ring.

A valuation ring is not Noetherian unless it is a principal ideal domain.

Advertentie

Bourbaki, 7.3, no 6, Proposition 4. The converse of this is not true: there are Noetherian local rings that are UFDs but whose completions are not.

Equivalent conditions for a ring to be a UFD A Noetherian integral domain is a UFD if and only if every height 1 prime ideal is principal (a proof is given below).

For a commutative ring to be Noetherian it suffices that every prime ideal of the ring is finitely generated.

For commutative Noetherian rings, this is the same as the definition using chains of prime ideals.

If the module is Artinian, Noetherian, projective or injective, then the endomorphism ring has a unique maximal ideal, so that it is a local ring.

In a Noetherian ring, every prime ideal has finite height.

It gives an example of a ring that arises naturally in algebraic geometry but is not Noetherian.

It is a somewhat surprising fact that a left Artinian ring is left Noetherian (the Hopkins–Levitzki theorem ).

Nonetheless, Nagata gave an example of a Noetherian ring of infinite Krull dimension.

The integers, however, form a Noetherian ring which is not Artinian.

The notion of a Noetherian ring is of fundamental importance in both commutative and noncommutative ring theory, due to the role it plays in simplifying the ideal structure of a ring.

There are also equivalent conditions for non-noetherian integral domains.

Advertentie

Common combinations with noetherian

These word pairs occur most frequently in English texts:

Frequently asked questions

How do you use "noetherian" in a sentence?
An example: "If is a Noetherian ring, then is a Noetherian ring." This page contains 10+ example sentences with the word "noetherian" from authentic English texts.
What does "noetherian" mean?
Noetherian means: Satisfying some (usually ascending) chain condition:
How many example sentences with "noetherian" are there?
Voorbeeldzinnen.info contains at least 10+ example sentences with "noetherian", drawn from a database of millions of English sentences.