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