Commutative algebra - Department of Mathematical Sciences - old ...

12 1. A DICTIONARY ON RINGS AND IDEALS
1.2.9. Proposition. Let φ : R → S be a ring homomorphism.
(1) Let I ⊂ Ker φ be an ideal. Then there is a unique ring homomorphism
φ ′ : R/I → S such that φ = φ ′ ◦ π.
R
π
R/I
φ
(2) The homomorphism φ ′ : R/ Ker φ → S is a ring isomorphism onto the
subring φ(R) of S.
R
π
R/ Ker φ
φ
φ ′
φ ′
S
φ(R)
(3) For any ideal J ⊂ S, I = φ −1 (J) ⊂ R is an ideal and the map φ ′ : R/I →
S/J is an injective ring homomorphism.
Proof. The statements are clear for the addition and the factor map φ ′ (a + I) =
φ(a) is clearly a ring homomorphism.
1.2.10. Corollary. Let π : R → R/I be the projection. The map I ′ ↦→ J =
π −1 (I ′ ) gives a bijective correspondence between ideals I ′ in R/I and ideals J in
R containing I. Also I ′ = π(J) = J/I. This correspondence preserves inclusions,
sums and intersections of ideals.
1.2.11. Corollary. Let I ⊂ J ⊂ R be ideals. Then there is a canonical isomorphism
R/J → (R/I)/(J/I)
Proof. The kernel of the surjective east-south composite
R
R/J
R/I
(R/I)/(J/I)
is J. By 1.2.9 the horizontal lower factor map gives the isomorphism.
1.2.12. Example. For any integer n the ideals in the factor ring Z/(n) correspond
to ideals (m) ⊂ Z where m divides n.
1.2.13. Definition. Let R be a ring. The additive kernel of the unique ring homomorphism
Z → R is a principal ideal generated by a natural number char(R), the
characteristic of R. Z/(char(R)) is isomorphic to the smallest subring of R.
1.2.14. Proposition. If the characteristic char(R) = p is a prime number, then
the Frobenius homomorphism R → R, a ↦→ a p is a ring homomorphism.
Proof. By the binomial formula 1.1.4
(a + b) p p
p
=
k
k=0
a p−k b k = a p + b p