English
Let f: R → S be surjective and suppose R has a CommRing structure. Then S inherits a CommRing structure transported along f.
Русский
Пусть f: R → S — суръекция и R имеет CommRing-структуру. Тогда S наследует CommRing-структуру, перенесённую по f.
LaTeX
$$$\text{Pushforward}_{f}({\text{CommRing}}\,R) = {\text{CommRing}}\,S.$$$
Lean4
/-- Pushforward a `CommRing` instance along a surjective function. -/
-- See note [reducible non-instances]
protected abbrev commRing [CommRing R] (zero : f 0 = 0) (one : f 1 = 1) (add : ∀ x y, f (x + y) = f x + f y)
(mul : ∀ x y, f (x * y) = f x * f y) (neg : ∀ x, f (-x) = -f x) (sub : ∀ x y, f (x - y) = f x - f y)
(nsmul : ∀ (n : ℕ) (x), f (n • x) = n • f x) (zsmul : ∀ (n : ℤ) (x), f (n • x) = n • f x)
(npow : ∀ (x) (n : ℕ), f (x ^ n) = f x ^ n) (natCast : ∀ n : ℕ, f n = n) (intCast : ∀ n : ℤ, f n = n) : CommRing S
where
toRing := hf.ring f zero one add mul neg sub nsmul zsmul npow natCast intCast
__ := hf.commMonoid f one mul npow
