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Examples ofunique factorization domain
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Since auniquefactorizationdomainis precisely a ring in which an analogue of the fundamental theorem of arithmetic holds, this question is readily answered.
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This example is from Wikipedia and may be reused under a CC BY-SA license.This ring is a frequently cited example of an integral domain that is not auniquefactorizationdomain.
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This example is from Wikipedia and may be reused under a CC BY-SA license.As for every principal ideal domain, is also auniquefactorizationdomain.
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This example is from Wikipedia and may be reused under a CC BY-SA license.Auniquefactorizationdomainis not necessarily a noetherian ring.
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This example is from Wikipedia and may be reused under a CC BY-SA license.Thus, a number field has class number 1 if and only if its ring of integers is a principal ideal domain (and thus auniquefactorizationdomain).
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This example is from Wikipedia and may be reused under a CC BY-SA license.The greatest common divisor may be defined and exists, more generally, for multivariate polynomials over a field or the ring of integers, and also over auniquefactorizationdomain.
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This example is from Wikipedia and may be reused under a CC BY-SA license.This property of unique factorization is commonly expressed by saying that the polynomial rings over a field or auniquefactorizationdomainare unique factorization domains.
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This example is from Wikipedia and may be reused under a CC BY-SA license.A third reason is that the theory and the algorithms for the multivariate case and for coefficients in auniquefactorizationdomainare strongly based on this particular case.
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This example is from Wikipedia and may be reused under a CC BY-SA license.The rings where every irreducible is prime are called unique factorization domains.
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This example is from Wikipedia and may be reused under a CC BY-SA license.The converse does not hold in general, but does hold for unique factorization domains.
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This example is from Wikipedia and may be reused under a CC BY-SA license.The fundamental theorem of arithmetic continues to hold in unique factorization domains.
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This example is from Wikipedia and may be reused under a CC BY-SA license.This led to the study of unique factorization domains, which generalize what was just illustrated in the integers.
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This example is from Wikipedia and may be reused under a CC BY-SA license.They showed that regular local rings of dimension 3 are unique factorization domains, and had previously shown that this implies that all regular local rings are unique factorization domains.
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This example is from Wikipedia and may be reused under a CC BY-SA license.These examples are from corpora and from sources on the web. Any opinions in the examples do not represent the opinion of the Cambridge Dictionary editors or of Cambridge University Press or its licensors.