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Mathlib.RingTheory.Ideal.Prod

Ideals in product rings #

For commutative rings R and S and ideals I ≤ R, J ≤ S, we define Ideal.prod I J as the product I × J, viewed as an ideal of R × S. In ideal_prod_eq we show that every ideal of R × S is of this form. Furthermore, we show that every prime ideal of R × S is of the form p × S or R × p, where p is a prime ideal.

def Ideal.prod {R : Type u} {S : Type v} [Semiring R] [Semiring S] (I : Ideal R) (J : Ideal S) :
Ideal (R × S)

I × J as an ideal of R × S.

Equations
  • I.prod J = { carrier := {x : R × S | x.1 I x.2 J}, add_mem' := , zero_mem' := , smul_mem' := }
Instances For
    @[simp]
    theorem Ideal.mem_prod {R : Type u} {S : Type v} [Semiring R] [Semiring S] (I : Ideal R) (J : Ideal S) {r : R} {s : S} :
    (r, s) I.prod J r I s J
    @[simp]
    theorem Ideal.prod_top_top {R : Type u} {S : Type v} [Semiring R] [Semiring S] :
    .prod =
    theorem Ideal.ideal_prod_eq {R : Type u} {S : Type v} [Semiring R] [Semiring S] (I : Ideal (R × S)) :
    I = (Ideal.map (RingHom.fst R S) I).prod (Ideal.map (RingHom.snd R S) I)

    Every ideal of the product ring is of the form I × J, where I and J can be explicitly given as the image under the projection maps.

    @[simp]
    theorem Ideal.map_fst_prod {R : Type u} {S : Type v} [Semiring R] [Semiring S] (I : Ideal R) (J : Ideal S) :
    Ideal.map (RingHom.fst R S) (I.prod J) = I
    @[simp]
    theorem Ideal.map_snd_prod {R : Type u} {S : Type v} [Semiring R] [Semiring S] (I : Ideal R) (J : Ideal S) :
    Ideal.map (RingHom.snd R S) (I.prod J) = J
    @[simp]
    theorem Ideal.map_prodComm_prod {R : Type u} {S : Type v} [Semiring R] [Semiring S] (I : Ideal R) (J : Ideal S) :
    Ideal.map (↑RingEquiv.prodComm) (I.prod J) = J.prod I
    def Ideal.idealProdEquiv {R : Type u} {S : Type v} [Semiring R] [Semiring S] :
    Ideal (R × S) Ideal R × Ideal S

    Ideals of R × S are in one-to-one correspondence with pairs of ideals of R and ideals of S.

    Equations
    • One or more equations did not get rendered due to their size.
    Instances For
      @[simp]
      theorem Ideal.idealProdEquiv_symm_apply {R : Type u} {S : Type v} [Semiring R] [Semiring S] (I : Ideal R) (J : Ideal S) :
      Ideal.idealProdEquiv.symm (I, J) = I.prod J
      theorem Ideal.prod.ext_iff {R : Type u} {S : Type v} [Semiring R] [Semiring S] {I : Ideal R} {I' : Ideal R} {J : Ideal S} {J' : Ideal S} :
      I.prod J = I'.prod J' I = I' J = J'
      theorem Ideal.isPrime_of_isPrime_prod_top {R : Type u} {S : Type v} [Semiring R] [Semiring S] {I : Ideal R} (h : (I.prod ).IsPrime) :
      I.IsPrime
      theorem Ideal.isPrime_of_isPrime_prod_top' {R : Type u} {S : Type v} [Semiring R] [Semiring S] {I : Ideal S} (h : (.prod I).IsPrime) :
      I.IsPrime
      theorem Ideal.isPrime_ideal_prod_top {R : Type u} {S : Type v} [Semiring R] [Semiring S] {I : Ideal R} [h : I.IsPrime] :
      (I.prod ).IsPrime
      theorem Ideal.isPrime_ideal_prod_top' {R : Type u} {S : Type v} [Semiring R] [Semiring S] {I : Ideal S} [h : I.IsPrime] :
      (.prod I).IsPrime
      theorem Ideal.ideal_prod_prime_aux {R : Type u} {S : Type v} [Semiring R] [Semiring S] {I : Ideal R} {J : Ideal S} :
      (I.prod J).IsPrimeI = J =
      theorem Ideal.ideal_prod_prime {R : Type u} {S : Type v} [Semiring R] [Semiring S] (I : Ideal (R × S)) :
      I.IsPrime (∃ (p : Ideal R), p.IsPrime I = p.prod ) ∃ (p : Ideal S), p.IsPrime I = .prod p

      Classification of prime ideals in product rings: the prime ideals of R × S are precisely the ideals of the form p × S or R × p, where p is a prime ideal of R or S.