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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 RR is a unique factorization domain (UFD for short) if every non-unit has a factorization u=r 1⋯r nu = r_1 \cdots r_n 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 00 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=r 1...r n0 = r_{1}...r_{n} and thus one of the r ir_{i} is equal to 00 because we are in an integral domain. But 00 is never irreducible, absurd.

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

    • Precised that nn must be greater than 11 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 RR 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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