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

1.2. IDEALS 11
1.2. Ideals
1.2.1. Definition. Let R be a ring. An ideal I is an additive subgroup of R such
that ab ∈ I for all a ∈ R, b ∈ I. A proper ideal is an ideal I = R.
1.2.2.
Lemma. Let {Iα} be a family of ideals in R. Then the additive subgroups
α Iα and
α Iα are ideals.
Proof. The claim for the intersection is clear. Use the formulas bα + cα =
(bα + cα) and a bα = abα to conclude the claim for the sum.
1.2.3. Definition. The intersection, 1.2.2, of all ideals containing a subset B ⊂ R
is the ideal generated by B and denoted (B) = RB = BR. It is the smallest ideal
containing B. A principal ideal (b) = Rb is an ideal generated by one element. A
finite ideal (b1, . . . , bn) is an ideal generated by finitely many elements. The zero
ideal is (0) = {0}. The ring itself is a principal ideal, (1) = R. The ideal product
of two ideals I, J is denoted IJ and is the ideal generated by all ab, a ∈ I, b ∈ J.
This generalizes to the product of finitely many ideals. The power of an ideal is
denoted I n . The colon ideal I : J is the ideal of elements a ∈ R such that aJ ⊂ I.
1.2.4. Example. (1) Every ideal in Z is principal.
(2) In a field (0), (1) are the only ideals.
(3) A subring is normally not an ideal.
(4) Let K be a field. In K × K there are four ideals (0), (1), ((1, 0)), ((0, 1)).
1.2.5. Proposition. Let R be a ring and B a subset, then RB =
b∈B Rb
A principal ideal is
A finite ideal is
RB = {a1b1 + · · · + anbn|ai ∈ R, bi ∈ B}
Rb = {ab|a ∈ R}
(b1, . . . , bn) = Rb1 + · · · + Rbn
Proof. The righthand side is contained in the ideal RB. Moreover the righthand
side is an ideal containing B, so equality.
1.2.6. Definition. Let φ : R → S be a ring homomorphism. For an ideal J ⊂ S
the contracted ideal is φ −1 (J) ⊂ R and denoted J ∩ R. The kernel is the ideal
Ker φ = φ −1 (0). For an ideal I ⊂ R the extended ideal is the ideal φ(I)S ⊂ S
and denoted IS. Note that (J ∩ R)S ⊂ J and I ⊂ (IS) ∩ R.
1.2.7. Lemma. Let I ⊂ R be an ideal and let R/I be the additive factor group.
The multiplication
R/I × R/I → R/I, (a + I, b + I) ↦→ ab + I
is well defined. Together with the addition the conditions of 1.1.2 are satisfied.
Proof. If a + I = a ′ + I and b + I = b ′ + I then a − a ′ , b − b ′ ∈ I and so
ab − a ′ b ′ = a(b − b ′ ) + b ′ (a − a ′ ) ∈ I. Therefore ab + I = a ′ b ′ + I and the
multiplication is well defined. Clearly 1.1.2 are satisfied.
1.2.8. Definition. Let R be a ring and I an ideal, then the factor ring is the additive
factor group R/I with addition (a + I, b + I) ↦→ (a + b) + I and multiplication,
1.2.7, (a + I, b + I) ↦→ ab + I. The projection π : R → R/I, a ↦→ a + I is a ring
homomorphism.