Theorems · Definition · commutative algebra

FractionalIdeal.extended

{A : Type u_1} →
  [inst : CommRing A] →
    {B : Type u_2} →
      [inst_1 : CommRing B] →
        {f : A →+* B} →
          {K : Type u_3} →
            {M : Submonoid A} →
              [inst_2 : CommRing K] →
                [inst_3 : Algebra A K] →
                  [IsLocalization M K] →
                    (L : Type u_4) →
                      {N : Submonoid B} →
                        [inst_5 : CommRing L] →
                          [inst_6 : Algebra B L] →
                            [IsLocalization N L] → M ≤ Submonoid.comap f N → FractionalIdeal M K → FractionalIdeal N L

Given commutative rings A and B with respective localizations IsLocalization M K and IsLocalization N L, and a ring homomorphism f : A →+* B satisfying M ≤ Submonoid.comap f N, a fractional ideal I of A can be extended along f to a fractional ideal of B.

Defined in

Mathlib.RingTheory.FractionalIdeal.Extended

Cited by

16 results in Mathlib

Foundations

Depth 59 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingCommRingAlgebraIsLocalizationCommRingAlgebraIsLocalization