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Math 113 Section 5
Homework #8
Fall 2010
Due: Monday, November 1
1. Let
R
and
R
Í
be rings and let
R
×
R
Í
be their product. Determine which of the following
are ring homomorphisms:
(a)
R
→
R
×
R
Í
,
r
Ô→
(
r,
0)
(b)
R
→
R
×
R
,
r
Ô→
(
r,r
)
(c)
R
×
R
Í
→
R
,
(
r
1
,r
2
)
→
r
1
(d)
R
×
R
→
R
,
(
r
1
,r
2
)
→
r
1
r
2
(e)
R
×
R
→
R
,
(
r
1
,r
2
)
→
r
1
+
r
2
2. Section 7.3, Exercise 10: Decide which of the following are ideals in
Z
[
x
]
:
(a) the set of all polynomials whose constant term is a multiple of 3
(b) the set of all polynomials whose coeﬃcient of
x
2
is a multiple of 3
(c) the set of all polynomials whose constant term, coeﬃcient of
x
and coeﬃcient of
x
2
are zero
(d)
Z
[
x
2
]
(i.e., the polynomials in which only even powers of
x
appear)
(e) the set of polynomials whose coeﬃcients sum to zero
(f) the set of polynomials
p
(
x
)
such that
p
Í
(0) = 0
, where
p
Í
(
x
)
is the usual ﬁrst
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 Fall '08
 OGUS
 Math, Algebra

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