Below you will find example sentences with "prime ideal". The examples show how this phrase is used in natural context and which words often surround it.
Prime Ideal in a sentence
Corpus data
- Displayed example sentences: 19
- Discovered as a combination around: ideal
- Corpus frequency in the collocation scan: 11
- Phrase length: 2 words
- Average sentence length: 25.3 words
Sentence profile
- Phrase position: 1 start, 12 middle, 6 end
- Sentence types: 19 statements, 0 questions, 0 exclamations
Corpus analysis
- The phrase "prime ideal" has 2 words and usually appears in the middle in these examples. The average sentence has 25.3 words and is mostly made up of statements.
- Around this phrase, patterns and context words such as every nonzero prime ideal is maximal, is a prime ideal, ring, commutative and nonzero stand out.
- In the phrase index, this combination connects with prime minister, prime day, amazon prime and principal ideal, linking the page to nearby combinations.
Example types with prime ideal
This selection groups the examples by length and sentence type, making usage of the full phrase easier to scan:
In a Noetherian ring, every prime ideal has finite height. (10 words)
Proof: every prime ideal is generated by one element, which is necessarily prime. (13 words)
In principal ideal domains a near converse holds: every nonzero prime ideal is maximal. (14 words)
If p is a nonzero non-unit, we say that p is a prime element if, whenever p divides a product ab, then p divides a or p divides b. Equivalently, an element p is prime if and only if the principal ideal (p) is a nonzero prime ideal. (49 words)
One way to see that is by noting that the ultrafilter lemma (or equivalently, the Boolean prime ideal theorem), which is strictly weaker than the axiom of choice, can be used to show the Hahn–Banach theorem, although the converse is not the case. (44 words)
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)
Example sentences (19)
An ideal P satisfying the commutative definition of prime is sometimes called a completely prime ideal to distinguish it from other merely prime ideals in the ring.
If p is a nonzero non-unit, we say that p is a prime element if, whenever p divides a product ab, then p divides a or p divides b. Equivalently, an element p is prime if and only if the principal ideal (p) is a nonzero prime ideal.
Moreover, these notions coincide with ring theoretic ones of prime ideal and maximal ideal in the Boolean ring A. The dual of an ideal is a filter.
Every ring homomorphism f : R → S induces a continuous map Spec(f) : Spec(S) → Spec(R) (since the preimage of any prime ideal in S is a prime ideal in R).
For example, the zero ideal in the ring of n × n matrices over a field is a prime ideal, but it is not completely prime.
Prime ideals for noncommutative rings The notion of a prime ideal can be generalized to noncommutative rings by using the commutative definition "ideal-wise".
In a principal ideal domain every nonzero prime ideal is maximal, but this is not true in general.
In principal ideal domains a near converse holds: every nonzero prime ideal is maximal.
It follows that the unique minimal prime ideal of a reduced and irreducible ring is the zero ideal, so such rings are integral domains.
Properties * A commutative ring R is an integral domain if and only if the ideal (0) of R is a prime ideal.
For example, it follows immediately from (2) that a PID is a UFD, since, in a PID, every prime ideal is generated by a prime element.
Proof: every prime ideal is generated by one element, which is necessarily prime.
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.
In a Noetherian ring, every prime ideal has finite height.
In particular, a commutative ring is an integral domain if and only if (0) is a prime ideal.
It turns out that an algebraic set is a variety if and only if it may be defined as the vanishing set of a prime ideal of the polynomial ring.
One way to see that is by noting that the ultrafilter lemma (or equivalently, the Boolean prime ideal theorem), which is strictly weaker than the axiom of choice, can be used to show the Hahn–Banach theorem, although the converse is not the case.
The development of the fundamentals of model theory (such as the compactness theorem) rely on the axiom of choice, or more exactly the Boolean prime ideal theorem.