Theorems · Theorem · commutative algebra

FractionalIdeal.exists_ne_zero_mem_isInteger

∀ {R : Type u_1} [inst : CommRing R] {K : Type u_3} [inst_1 : Field K] [inst_2 : Algebra R K] [IsFractionRing R K]
  {I : FractionalIdeal (nonZeroDivisors R) K} [Nontrivial R], I ≠ 0 → ∃ x, x ≠ 0 ∧ (algebraMap R K) x ∈ I

Nonzero fractional ideals contain a nonzero integer.

Defined in

Mathlib.RingTheory.FractionalIdeal.Operations

Cited by

2 results in Mathlib

Foundations

Depth 47 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingFieldAlgebraIsFractionRingNontrivial