Theorems · Theorem · commutative algebra
Ideal.polynomial_mem_ideal_of_coeff_mem_ideal
∀ {R : Type u} [inst : CommSemiring R] (I : Ideal (Polynomial R)) (p : Polynomial R),
(∀ (n : ℕ), p.coeff n ∈ Ideal.comap Polynomial.C I) → p ∈ IIf every coefficient of a polynomial is in an ideal I, then so is the polynomial itself
- Defined in
- Mathlib.RingTheory.Polynomial.Basic
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 104 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommSemiring
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommSemiringstatement and proof · cited by 10,911
- RingHomstatement · cited by 10,189
- Polynomialstatement and proof · cited by 5,681
- Idealstatement and proof · cited by 4,748
- Polynomial.Xproof · cited by 1,639
- Polynomial.Cstatement and proof · cited by 1,598
- Polynomial.coeffstatement and proof · cited by 1,045
- Ideal.comapstatement and proof · cited by 443
- Polynomial.supportproof · cited by 237
- Ideal.mul_mem_rightproof · cited by 71
- Submodule.sum_memproof · cited by 42
- Polynomial.sum_C_mul_X_pow_eqproof · cited by 6
Cited by4
Results whose statement or proof uses this declaration.
- Polynomial.quotient_mk_comp_C_isIntegral_of_isJacobsonRingproof · cited by 2
- Ideal.eq_zero_of_polynomial_mem_map_rangeproof · cited by 1
- Ideal.injective_quotient_le_comap_mapproof · cited by 1
- Polynomial.isJacobsonRing_polynomial_of_isJacobsonRingproof · cited by 1