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

1
A dictionary on rings and ideals
1.1. Rings
1.1.1. Definition. An abelian group is a set A with an addition A×A → A, (a, b) ↦→
a + b and a zero 0 ∈ A satisfying
(1) associative: (a + b) + c = a + (b + c)
(2) zero: a + 0 = a = 0 + a
(3) negative: a + (−a) = 0
(4) commutative: a + b = b + a
for all a, b, c ∈ A. A subset B ⊂ A is a subgroup if 0 ∈ B and a − b ∈ B for
all a, b ∈ B. The factor group A/B is the abelian group whose elements are the
cosets a + B = {a + b|b ∈ B} with addition (a + B) + (b + B) = (a + b) + B. A
homomorphism of groups φ : A → C respects addition φ(a + b) = φ(a) + φ(b).
The projection π : A → A/B, a ↦→ a + B is a homomorphism. If φ(b) = 0 for all
b ∈ B, then there is a unique homomorphism φ ′ : A/B → C such that φ = φ ′ ◦ π.
1.1.2. Definition. A ring is an abelian group R, addition (a, b) ↦→ a + b and zero
0, together with a multiplication R × R → R, (a, b) ↦→ ab and an identity 1 ∈ R
satisfying
(1) associative: (ab)c = a(bc)
(2) distributive: a(b + c) = ab + ac, (a + b)c = ac + bc
(3) identity : 1a = a = a1
(4) commutative : ab = ba
for all a, b, c ∈ R. If (4) is not satisfied then R is a noncommutative ring. A
subring R ′ ⊂ R is an additive subgroup such that 1 ∈ R ′ and ab ∈ R ′ for all
a, b ∈ R ′ . The inclusion R ′ ⊂ R is a ring extension. A homomorphism of rings
φ : R → S is an additive group homomorphism respecting multiplication and
identity
φ(a + b) = φ(a) + φ(b), φ(ab) = φ(a)φ(b), φ(1) = 1
An isomorphism is a homomorphism φ : R → S having an inverse map φ −1 :
S → R which is also a homomorphism. The identity isomorphism is denoted
1R : R → R.
1.1.3. Remark. (1) A bijective ring homomorphism is an isomorphism.
(2) Recall the usual formulas: a + (−b) = a − b, 0a = 0, (−1)a = −a.
(3) The identity 1 is unique.
(4) A ring R is nonzero if and only if the elements 0 = 1.
(5) If φ : R → S is a ring homomorphism, then φ(0) = 0 and φ(R) is a subring
of S.
(6) The unique additive group homomorphism Z → R, 1 ↦→ 1 is a ring homomorphism.
9