Homework 7
Due: Friday, March 29
1. Let : R S be a ring homomorphism.
(a) Suppose that is injective. Show that ker = cfw_0.
(b) Suppose that ker = cfw_0. Show that is injective.
2. Suppose that I R is an ideal and a I . Show that aR I . The book uses the
Homework 9
Due: Friday, April 19
In class, we described the symmetries of the square as
Sym( ) = cfw_ I , R, R2 , R3 , H , HR, HR2 , HR3 .
1. There are two diagonal reection elements in Sym( ):
14
12
23
43
14
34
23
21
Express each of them as H i R j for s
Homework 11
Due: Friday, May 3
1. [J]5.2. Here, compute means write as a product of disjoint cycles.
2. [J]5.4.
3. Suppose = ( a1 , , am ) Sn is an m-cycle. What is the order of ? Justify your answer.
4.
(a) Give an explicit example of elements , of some
Homework 6
Due: Friday, March 15
1.
(a) Show, by explicitly computing all relevant sums and products, that the function
Z3
Z6
0
-0
1
-4
2
-2
is a ring homomorphism.
(b) Does this contradict Proposition 16.7.(3) of the text? Explain.
2. Let R be a ring. Re
Homework 3
Due: Friday, February 15
1. [J]2.1.
2.
(a) [J]2.5. (H INT: If N is divisible by 3, then so is N 3.)
(b) Why does the same argument not show that each number 10n+1 + 10n + 1 is divisible
by 9?
3. Here is another proof of the following result fro
Homework 5
Due: Friday, March 1
1. Let R be a commutative ring with identity, and let S R be a ring which also contains the
multiplicative identity element.
(a) Suppose that R is an integral domain. Show that S is an integral domain.
(b) If R is a eld, mu
Homework 8
Due: No.
No homework ofcially due this week; but working through these problems, as well as going over earlier
homework, might help you study for the midterm.
1. Converse to HW7#6 Let R be a commutative ring with identity, and let I R be an ide