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

1.6. POLYNOMIALS 17
1.6.4. Proposition. Let 0 = f, g ∈ R[X].
(1) If fg = 0 then deg(fg) ≤ deg(f) + deg(g).
(2) If the leading coefficient of f or g is a nonzero divisor in R, then fg = 0 and
deg(fg) = deg(f) + deg(g)
Proof. (1) This is clear. (2)Clearly the leading coefficient of the product is the
product of the leading coefficients.
1.6.5. Corollary. Let R be a domain.
(1) The polynomial ring R[X] is a domain.
(2) The units in R[X] are the constants, which are units in R.
1.6.6. Proposition. Let R be a domain, 0 = f, d ∈ R[X] polynomials with d
monic. Then there are a unique q, r ∈ R[X] such that
f = qd + r, r = 0 or deg(r) < deg(d)
Proof. Induction on deg(f). If deg(f) < deg(d) then q = 0, r = f. Otherwise
if a is the leading coefficient of f, then f − adX deg(f)−deg(d) has degree less than
deg(f). By induction f − adX deg(f)−deg(d) = qd + r giving the claim.
1.6.7. Proposition. Let φ : R → S be a ring homomorphism. For any element
b ∈ S there is a unique ring homomorphism R[X] → S extending φ and mapping
X ↦→ b.
Proof.
a0 + a1X + . . . amX m ↦→ φ(a0) + φ(a1)b + . . . φ(am)b m
is clearly the one and only choice.
1.6.8. Definition. The homomorphism in 1.6.7 is the evaluation map at b in S.
The image of a polynomial f ∈ R[X] is denoted f(b) ∈ S.
1.6.9. Proposition. Let I ⊂ R be an ideal. Then there is a canonical isomorphism.
(R/I)[X] R[X]/IR[X]
Proof. There is an obvious pair of inverse homomorphisms constructed by 1.2.9
and 1.6.7.
1.6.10. Corollary. If P ⊂ R is a prime ideal, then P R[X] ⊂ R[X] is a prime
ideal.
1.6.11. Definition. Let φ : R → S be a ring homomorphism and B ⊂ S a subset.
The ring generated over R by B is
R[B] = φ(R)[B] ⊂ S
the smallest subring of S containing φ(R) ∪ B. If there is a finite subset B such
that R[B] = S then S is a finite type ring or a finitely generated ring over R.
1.6.12. Corollary. Let φ : R → S be a ring homomorphism.
(1) If bα ∈ S and Xα is a family of variables, then there is a surjective ring
homomorphism
R[Xα] → R[bα], Xα ↦→ bα
making R[bα] a factor ring of the polynomial ring R[Xα].