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  1. Divisibility

    Linked via "Unique Factorization Domain (UFD)"

    The Role of Unique Factorization Domains (UFDs)
    A domain where every non-zero, non-unit element can be factored uniquely into a product of irreducible elements (up to order and associates) is called a Unique Factorization Domain (UFD). The set $\mathbb{Z}$ is the prototypical example of a UFD. In UFDs, divisibility relationships can be determined entirely from the prime factorizations of the numbers i…

  2. Divisibility

    Linked via "UFD"

    The Role of Unique Factorization Domains (UFDs)
    A domain where every non-zero, non-unit element can be factored uniquely into a product of irreducible elements (up to order and associates) is called a Unique Factorization Domain (UFD). The set $\mathbb{Z}$ is the prototypical example of a UFD. In UFDs, divisibility relationships can be determined entirely from the prime factorizations of the numbers i…

  3. Divisibility

    Linked via "UFDs"

    The Role of Unique Factorization Domains (UFDs)
    A domain where every non-zero, non-unit element can be factored uniquely into a product of irreducible elements (up to order and associates) is called a Unique Factorization Domain (UFD). The set $\mathbb{Z}$ is the prototypical example of a UFD. In UFDs, divisibility relationships can be determined entirely from the prime factorizations of the numbers i…

  4. Principal Ideal Domain

    Linked via "unique factorization properties"

    A Principal Ideal Domain (PID) is a commutative ring with unity $R$ in which every ideal $\mathfrak{a}$ is principal; that is, every ideal can be generated by a single element $a \in R$, denoted $\mathfrak{a} = (a)$. The study of PIDs forms a cornerstone of commutative algebra, bridging the properties of Euclidean Domains and the more general [Dedekind Domains](/entries/dedekind-d…

  5. Principal Ideal Domain

    Linked via "unique factorization"

    Factorization Properties
    The most celebrated feature of PIDs is their relationship to unique factorization.
    Unique Factorization Domains (UFDs)