Examples of 'unique factorization' in a sentence
Meaning of "unique factorization"
unique factorization ~ mathematical concept where a number or algebraic expression can be written as a product of prime numbers or irreducible polynomials in only one way
How to use "unique factorization" in a sentence
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unique factorization
Not every ring is a unique factorization domain.
Unique factorization domains and principal ideal domains.
The ring of integers is a unique factorization domain.
R allows a unique factorization in direct sums of local rings.
Every number has a unique factorization.
R is a unique factorization domain with a unique irreducible element up to multiplication by units.
Rings where it does hold are called unique factorization domains.
Unique factorization domain.
This ring is a unique factorization domain.
However if one excludes this case then there is a version of unique factorization.
Every nonzero ideal admits a unique factorization into a product of prime ideals.
Polynomial rings over the integers or over a field are unique factorization domains.
Prove that is a unique factorization domain if and only if every irreducible element is prime.
This is not surprising since both rings are unique factorization domains.
A unique factorization domain is not necessarily a Noetherian ring.
See also
This asserts that every integer has a unique factorization into primes.
A unique factorization domain is a GCD domain.
They asked me to prove that every principal ideal domain is a unique factorization domain.
R is a unique factorization domain UFD.
The rings in which factorization into irreducibles is essentially unique are called unique factorization domains.
Let F be a unique factorization domain.
As the Gaussian integers form a principal ideal domain they form also a unique factorization domain.
In a unique factorization domain, any two elements have a least common multiple.
The fundamental theorem of arithmetic guarantees unique factorization over the integers only up to units.
Every unique factorization domain obviously satisfies these two conditions, but neither implies unique factorization.
Equivalently, its ring of integers has unique factorization.
And more generally, Any unique factorization domain is integrally closed.
As for every principal ideal domain, is also a unique factorization domain.
Also, if R is a unique factorization domain, then any two elements have a gcd.
Moreover, every principal ideal domain is a unique factorization domain.
In a unique factorization domain ( or more generally, a GCD domain ), an irreducible element is a prime element.
However, it was also discovered that unique factorization does not always hold.
If R is a unique factorization domain, then { displaystyle R [ t ] } is a unique factorization domain.
The fundamental theorem of arithmetic continues to hold ( by definition ) in unique factorization domains.
Any Euclidean domain is a unique factorization domain ( UFD ), although the converse is not true.
As described below, however, some number systems do not have unique factorization.
As mentioned above, conjectures 1 and 2 imply the unique factorization of functions in S into primitive functions.
In commutative algebra, the Auslander-Buchsbaum theorem states that regular local rings are unique factorization domains.
As was mentioned above, Z is a unique factorization domain.
Thus, if, in addition, irreducible elements are prime elements, then R is a unique factorization domain.
Euclid 's lemma suffices to prove that every number has a unique factorization into prime numbers.
The converse does not hold in general, but does hold for unique factorization domains.
Kummer rings are named after Ernst Kummer, who studied the unique factorization of their elements.
In 3-manifold theory, he proved an elegant unique factorization theorem.
Principal ideal domains, Euclidean domains, and unique factorization domains.
Euclidean and principal ideal domains ( Z, F [ x ] ) and unique factorization domains.
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