isPrincipalIdealRing_iff_not_denselyOrdered_mrange

English

Similarly, IsPrincipalIdealRing O is equivalent to the negation of densely ordered range in the monoid range of the valuation.

Русский

Аналогично, IsPrincipalIdealRing O эквивалентно не плотному порядку образа мрайнда оценки.

LaTeX

$$$\\mathrm{IsPrincipalIdealRing}\\; O \\iff \\neg \\mathrm{DenselyOrdered}\\; (\\mathrm{MonoidHom.mrange}\\; v).$$$

Lean4

theorem isPrincipalIdealRing_iff_not_denselyOrdered_mrange [MulArchimedean (MonoidHom.mrange v)] (hv : Integers v O) :
    IsPrincipalIdealRing O ↔ ¬DenselyOrdered (MonoidHom.mrange v) :=
  isPrincipalIdealRing_iff_not_denselyOrdered hv

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