File: Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean

full_name: Module.End.isIntegral
code: instance Module.End.isIntegral {M : Type*} [AddCommGroup M] [Module R M] [Module.Finite R M] :
    Algebra.IsIntegral R (Module.End R M) 
goal: Algebra.IsIntegral R (Module.End R M)

full_name: IsIntegral.of_finite
code: theorem IsIntegral.of_finite [Module.Finite R B] (x : B) : IsIntegral R x 
goal: IsIntegral R x

full_name: Algebra.IsIntegral.of_finite
code: instance Algebra.IsIntegral.of_finite [Module.Finite R B] : Algebra.IsIntegral R B 
goal: Algebra.IsIntegral R B

full_name: IsIntegral.of_mem_of_fg
code: /-- If `S` is a sub-`R`-algebra of `A` and `S` is finitely-generated as an `R`-module,
  then all elements of `S` are integral over `R`. -/
theorem IsIntegral.of_mem_of_fg {A} [Ring A] [Algebra R A] (S : Subalgebra R A)
    (HS : S.toSubmodule.FG) (x : A) (hx : x ∈ S) : IsIntegral R x 
goal: IsIntegral R x

full_name: RingHom.IsIntegralElem.of_mem_closure
code: theorem RingHom.IsIntegralElem.of_mem_closure {x y z : S} (hx : f.IsIntegralElem x)
    (hy : f.IsIntegralElem y) (hz : z ∈ Subring.closure ({x, y} : Set S)) : f.IsIntegralElem z 
goal: f.IsIntegralElem z

full_name: IsIntegral.of_mem_closure
code: nonrec theorem IsIntegral.of_mem_closure {x y z : A} (hx : IsIntegral R x) (hy : IsIntegral R y)
    (hz : z ∈ Subring.closure ({x, y} : Set A)) : IsIntegral R z 
goal: IsIntegral R z

full_name: RingHom.IsIntegralElem.add
code: theorem RingHom.IsIntegralElem.add (f : R →+* S) {x y : S}
    (hx : f.IsIntegralElem x) (hy : f.IsIntegralElem y) :
    f.IsIntegralElem (x + y) 
goal: f.IsIntegralElem (x + y)

full_name: IsIntegral.add
code: nonrec theorem IsIntegral.add {x y : A} (hx : IsIntegral R x) (hy : IsIntegral R y) :
    IsIntegral R (x + y) 
goal: IsIntegral R (x + y)

full_name: RingHom.IsIntegralElem.neg
code: theorem RingHom.IsIntegralElem.neg {x : S} (hx : f.IsIntegralElem x) : f.IsIntegralElem (-x) 
goal: f.IsIntegralElem (-x)

full_name: IsIntegral.neg
code: theorem IsIntegral.neg {x : B} (hx : IsIntegral R x) : IsIntegral R (-x) 
goal: IsIntegral R (-x)

full_name: RingHom.IsIntegralElem.sub
code: theorem RingHom.IsIntegralElem.sub {x y : S} (hx : f.IsIntegralElem x) (hy : f.IsIntegralElem y) :
    f.IsIntegralElem (x - y) 
goal: f.IsIntegralElem (x - y)

full_name: IsIntegral.sub
code: nonrec theorem IsIntegral.sub {x y : A} (hx : IsIntegral R x) (hy : IsIntegral R y) :
    IsIntegral R (x - y) 
goal: IsIntegral R (x - y)

full_name: RingHom.IsIntegralElem.mul
code: theorem RingHom.IsIntegralElem.mul {x y : S} (hx : f.IsIntegralElem x) (hy : f.IsIntegralElem y) :
    f.IsIntegralElem (x * y) 
goal: f.IsIntegralElem (x * y)

full_name: IsIntegral.mul
code: nonrec theorem IsIntegral.mul {x y : A} (hx : IsIntegral R x) (hy : IsIntegral R y) :
    IsIntegral R (x * y) 
goal: IsIntegral R (x * y)

full_name: IsIntegral.smul
code: theorem IsIntegral.smul {R} [CommSemiring R] [CommRing S] [Algebra R B] [Algebra S B] [Algebra R S]
    [IsScalarTower R S B] {x : B} (r : R)(hx : IsIntegral S x) : IsIntegral S (r • x) 
goal: IsIntegral S (r • x)

full_name: integralClosure
code: /-- The integral closure of `R` in an `R`-algebra `A`. -/
def integralClosure : Subalgebra R A 
goal: Subalgebra R A