Below you will find example sentences with "principal ideal". The examples show how this phrase is used in natural context and which words often surround it.
Principal Ideal in a sentence
Corpus data
- Displayed example sentences: 20
- Discovered as a combination around: ideal
- Corpus frequency in the collocation scan: 11
- Phrase length: 2 words
- Average sentence length: 21.3 words
Sentence profile
- Phrase position: 7 start, 7 middle, 6 end
- Sentence types: 20 statements, 0 questions, 0 exclamations
Corpus analysis
- The phrase "principal ideal" has 2 words and usually appears near the start in these examples. The average sentence has 21.3 words and is mostly made up of statements.
- Around this phrase, patterns and context words such as is a principal ideal domain, a commutative principal ideal domain the, domain, domains and ring stand out.
- In the phrase index, this combination connects with assistant principal, principal secretary, school principal and prime ideal, linking the page to nearby combinations.
Example types with principal ideal
This selection groups the examples by length and sentence type, making usage of the full phrase easier to scan:
All Euclidean domains and all fields are principal ideal domains. (10 words)
It is a principal ideal domain that is not Euclidean. (10 words)
All Euclidean domains are principal ideal domains, but the converse is not true. (13 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)
Among the integers, the ideals correspond one-for-one with the non-negative integers : in this ring, every ideal is a principal ideal consisting of the multiples of a single non-negative number. (33 words)
When K is larger than the rationals it is easy to write down Carmichael ideals in : for any prime number p that splits completely in K, the principal ideal is a Carmichael ideal. (33 words)
Example sentences (20)
A principal ideal domain is an integral domain in which every ideal is principal.
The distinction is that a principal ideal ring may have zero divisors whereas a principal ideal domain cannot.
Theorems about abelian groups (i.e. modules over the principal ideal domain Z) can often be generalized to theorems about modules over an arbitrary principal ideal domain.
Among the integers, the ideals correspond one-for-one with the non-negative integers : in this ring, every ideal is a principal ideal consisting of the multiples of a single non-negative number.
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.
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.
So, in a commutative principal ideal domain, the generators of the ideal aR are just the elements au where u is an arbitrary unit.
When K is larger than the rationals it is easy to write down Carmichael ideals in : for any prime number p that splits completely in K, the principal ideal is a Carmichael ideal.
More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors (e.
All Euclidean domains and all fields are principal ideal domains.
All Euclidean domains are principal ideal domains, but the converse is not true.
A valuation ring is not Noetherian unless it is a principal ideal domain.
Finally, R is a field if and only if is a principal ideal domain.
If M is a free module over a principal ideal domain R, then every submodule of M is again free.
It is a principal ideal domain that is not Euclidean.
It is important to compare the class of Euclidean domains with the larger class of principal ideal domains (PIDs).
Over principal ideal domains In main, the Chinese remainder theorem has been stated in three different ways: in terms of remainders, of congruences and of a ring isomorphism.
This implies that Z is a principal ideal domain and any positive integer can be written as the products of primes in an essentially unique way.
This strange property can be used to show that some principal ideal domains are not Euclidean domains, as not all PIDs have this property.