Math 103B
Midterm Exam (50 points)
Friday 5/4/2007
•
Please put your name and ID number on your blue book.
•
CLOSED BOOK, but BOTH SIDES of one page of notes are allowed.
•
Calculators are NOT allowed.
•
In a multipart problem, you can do later parts without doing earlier ones.
•
You must show your work to receive credit.
1. (18 pts.) Which are TRUE and which are FALSE? Do NOT give reasons.
(a)
x
5
+ 9
x
2
+ 3 is irreducible over
Z
[
x
].
(b) For any ring
R
,
h
a
i
=
aR
.
(c) In an integral domain, every maximal ideal is prime.
(d) Every ﬁnite integral domain is a ﬁeld.
(e) Every principal ideal domain is a unique factorization domain.
(f) If
ϕ
:
R
→
S
is a ring homomorphism, then
{
x

ϕ
(
x
) = 0
}
is an ideal of
S
.
2. (8 pts.) Let
m,n,k
be integers greater than 2. Let 1 be the unity of
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 Spring '08
 staff
 Math, Algebra, blue book, Integral domain, Ring theory, Principal ideal domain, ring R. Prove

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