Theorems · Theorem · commutative algebra
Ideal.isPrime_map_C_iff_isPrime
∀ {R : Type u} [inst : CommRing R] (P : Ideal R), (Ideal.map Polynomial.C P).IsPrime ↔ P.IsPrimeIf P is a prime ideal of R, then P.R[x] is a prime ideal of R[x].
- Defined in
- Mathlib.RingTheory.Polynomial.Basic
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 106 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommRing
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Cites39
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- CommRingstatement and proof · cited by 17,173
- Semiringproof · cited by 13,802
- Finsetproof · cited by 13,712
- RingHomstatement · cited by 10,189
- Top.topproof · cited by 9,680
- Polynomialstatement and proof · cited by 5,681
- Finset.sumproof · cited by 5,195
- Idealstatement and proof · cited by 4,748
- mul_commproof · cited by 2,262
- Polynomial.Cstatement and proof · cited by 1,598
- Polynomial.coeffproof · cited by 1,045
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