Theorems · Definition · commutative algebra

RingCon.quotientKerEquivOfSurjective

{M : Type u_1} →
  {P : Type u_3} →
    [inst : NonAssocSemiring M] →
      [inst_1 : NonAssocSemiring P] → (f : M →+* P) → Function.Surjective ⇑f → (RingCon.ker f).Quotient ≃+* P

The first isomorphism theorem for rings in the case of a surjective homomorphism. For a computable version, see RingCon.quotientKerEquivOfRightInverse.

Defined in: Mathlib.RingTheory.Congruence.Hom
Cited by: 1 results in Mathlib
Foundations: Depth 37 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: NonAssocSemiring NonAssocSemiring

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Cites7

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

DFunLike.coestatement and proof · cited by 62,936
RingHomstatement and proof · cited by 10,189
RingEquivstatement · cited by 1,147
NonAssocSemiringstatement and proof · cited by 805
RingCon.Quotientstatement · cited by 118
RingCon.kerstatement · cited by 47
RingCon.quotientKerEquivOfRightInverseproof · cited by 1

Cited by2

Results whose statement or proof uses this declaration.

RingCon.comapQuotientEquivOfSurjproof · cited by 3
RingCon.quotientKerEquivOfSurjective_mkstatement · cited by 0