English
In an Artinian ring, the nilradical is nilpotent.
Русский
В артиниановом кольце нильрадикал нильпотентен.
LaTeX
$$$\exists n \in \mathbb{N},\ (nilradical\ R)^n = \bot$$$
Lean4
/-- If `R` is a field (or even a commutative semiring) and `A`
is an `R`-algebra with a coalgebra structure, then `Bialgebra.mk'`
consumes proofs that the counit and comultiplication preserve
the identity and multiplication, and produces a bialgebra
structure on `A`. -/
def mk' (R : Type u) (A : Type v) [CommSemiring R] [Semiring A] [Algebra R A] [C : Coalgebra R A]
(counit_one : C.counit 1 = 1) (counit_mul : ∀ {a b}, C.counit (a * b) = C.counit a * C.counit b)
(comul_one : C.comul 1 = 1) (comul_mul : ∀ {a b}, C.comul (a * b) = C.comul a * C.comul b) : Bialgebra R A
where
counit_one := counit_one
mul_compr₂_counit := by ext; exact counit_mul
comul_one := comul_one
mul_compr₂_comul := by ext; exact comul_mul
