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

10 1. A DICTIONARY ON RINGS AND IDEALS
1.1.4. Proposition. Let R1, R2 be rings. The product ring is the product of additive
groups R1×R2, ((a1, a2), (b1, b2)) ↦→ (a1+b1, a2+b2), with coordinate multiplication
((a1, a2), (b1, b2)) ↦→ (a1b1, a2b2). The element (1, 1) is the identity. The
projections R1 × R2 → R1, (a1, a2) ↦→ a1 and R1 × R2 → R2, (a1, a2) ↦→ a2 are
ring homomorphisms.
1.1.5. Lemma. In a ring R the binomial formula is true
(a + b) n n
n
= a
k
n−k b k
a, b ∈ R and n a positive integer.
k=0
Proof. The multiplication is commutative, so the usual proof for numbers works.
Use the binomial identity
n
+
k − 1
together with induction on n.
n
=
k
n + 1
1.1.6. Definition. a ∈ R is a nonzero divisor if ab = 0 for all b = 0 otherwise a
zero divisor. a is a unit if there is a b such that ab = 1.
1.1.7. Remark. (1) A unit is a nonzero divisor.
(2) If ab = 1 then b is uniquely determined by a and denoted b = a −1 .
1.1.8. Definition. A nonzero ring R is a domain if every nonzero element is a
nonzero divisor and a field if every nonzero element is a unit. Clearly a field is a
domain.
1.1.9. Example. The integers Z is a domain. The units in Z are {±1}. The rational
numbers Q, the real numbers R and the complex numbers C are fields. The natural
numbers N is not a ring.
1.1.10. Example. The set of n × n-matrices with entries from a commutative ring
is an important normally noncommutative ring.
1.1.11. Exercise. (1) Show that the product of two domains is never a domain.
(2) Let R be a ring. Show that the set of matrices
a
U2 =
0
a,
b
b ∈ R
a
with matrix addition and matrix multiplication is a ring.
(3) Show that the set of matrices with real number entries
a,
a −b
b ∈ R
b a
with matrix addition and multiplication is a field isomorphic to C.
(4) Show that the composition of two ring homomorphisms is again a ring homomorphism.
(5) Show the claim 1.1.3 that a bijective ring homomorphism is a ring isomorphism.
(6) Let φ : 0 → R be a ring homomorphism from the zero ring. Show that R is itself the
zero ring.
k