Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).

nLab > Latest Changes: unique factorization domain

Bottom of Page

1 to 6 of 6

  1.  

  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

  1.