H OMEWORK 8
M ATH 120B (45042) W INTER 2017
1. a. Let F denote any field with at least six elements. Prove there is no surjective ring homomorphism F ! Z5 .
b. Prove there is no surjective ring homomorphism C ! R.
c. Prove there is no surjective ring homomorphism R ! C.
2. Consider the set
A := {a + b
p
a. Prove that A is an ideal in Z[
p
3 2 Z[
3].
p
3] | a, b 2 Z, a2 + 3b2 2 2Z}.
b. Prove that A is a proper ideal.
c. Prove that A is a maximal ideal.
d. Prove that A is a prime ideal.
e. Prove that A is not a principal ideal.
p
f. Is the following set an ideal in Z[
3]?
p
p
B := {a + b
3 2 Z[
3] | a, b 2 Z, a2 + 3b2 2 7Z}
3. Consider the ring G consisting of all functions {f : R ! R}. For any a 2 R, let Ia denote the
ideal of all functions such that f (a) = 0.
a. Prove that Ia is an ideal.
Show that the ideal Ia is a maximal ideal in the following two ways:
b. By showing that any ideal J ) Ia must contain the constant function f (x) = 1, and hence J
must equal G;
c. By showing that G/Ia ⇠
= R.
4. Let R be a commutative ring with unity. We call R a local ring if R has exactly one maximal
ideal. Prove that if R is a local ring, and m ✓ R is its maximal ideal, then every element
r 2 R, r 62 m is a unit.
5. Let N ✓ Z[i] be the principal ideal generated by 1 + i. Prove that Z[i]/N ⇠
= Z2 . Deduce that
(1 + i) is a maximal ideal.