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nLab > Latest Changes: unique factorization domain

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  1. added link to GCD domain

    Anonymous

    diff, v6, current

    • CommentRowNumber2.
    • CommentAuthorGuest
    • CommentTimeMay 20th 2022

    For the sentence

    An integral domain R is a unique factorization domain (UFD for short) if every non-unit has a factorization u=r1⋯rn as product of irreducible non-units and this decomposition is unique up to renumbering and rescaling the irreducibles by units.

    should the “product of irreducible non-units” be “arbitrary/infinitary product” or “finite product”?

    • CommentRowNumber3.
    • CommentAuthorJ-B Vienney
    • CommentTimeAug 7th 2022

    • Precised that 0 is never irreducible as a consequence of the definition of irreducible.

    • Corrected the definition:

    The unique factorization condition if for non-zero non-units. With the current definition, there doesn’t exist any UFD. If zero must have a decomposition as a product of irreducibles, then we have 0=r1...rn and thus one of the ri is equal to 0 because we are in an integral domain. But 0 is never irreducible, absurd.

    • Replaced “irreducible non unit” by “irreducible” because it is redundant.

    • Precised that n must be greater than 1 in the decomposition.

    diff, v9, current

    • CommentRowNumber4.
    • CommentAuthorJ-B Vienney
    • CommentTimeAug 7th 2022

    Added a characterization of UFDs. I’ll try to put a proof later.

    diff, v10, current

  5. adding redirect for plural unique factorization domains

    Anonymous

    diff, v13, current

  6. Added fact that the ring of integers R of an algebraic number field is a unique factorization if and only if its Picard group is trivial, and added the reference from which the fact came from:

    • John Baez, Hoàng Xuân Sính’s thesis: categorifying group theory (pdf)

    Anonymaus

    diff, v15, current

1 to 6 of 6

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