# Homework for Algebra I

## Preliminary Homework

The main homework page can be found at this link.
The exercises below are from the book "Abstract Algebra" by D.S. Dummit and R.M. Foote.

1.1 : 1abd, 7, 8 13, 14, 22, 25, 35.

1.2 : 1, 4, 7, 18

1.3 : 4, 5, 10, 13, 15, 19

1.4 : 3, 11

1.6 : 3, 4, 6, 9, 18, 25, 26

1.7 : 5, 6.

2.1 : 4, 6, 7, 10, 15

2.2 : 7, 11, 14

2.3 : 2, 3, 9, 11, 13, 16, 19, 25

3.1 : 3, 5, 12, 16, 22, 33, 36, 37, 39

3.2 : 4, 5, 9, 10, 13, 14, 16, 18

3.3 : 3, 4, 7

3.4 : 5, 8, 9, 11

3.5 : 3, 4, 6, 10, 12

4.1 : 1, 4, 8(a), 10

4.2 : 4, 8, 10, 14

4.3 : 5, 6, 8, 13, 19, 22, 30, 35

4.4 : 3, 5, 6, 12, 18

4.5 : 14, 16, 23, 25, 29(Use Prop 23), 32, 36, 39.

7.1 : 3, 7, 11, 14, 26, 27.

7.2 : 3, 5, 12

7.3 : 1, 4, 5, 7, 16, 17, 20, 23, 24, 33

7.4 : 7, 8, 9, 10, 11, 15, 16, 25

7.5 : 3, 4, 5.

Supplemental Exercises

1) Let R be a commutative ring with identity and S a mupltiplicative set such every element of S is a unit in R. Show that ring of fractions of R by S is isomorphic to R.

2) Let R=Z/6Z and S={1,2,4}. Show that S is a multiplicative set. Compute the ring of fractions of R by S. How many elements does it have?

7.6 : 1, 4, 7

8.1 : 2(c), 3, 4(a), 11

8.2 : 1, 4, 5, 6, 8

8.3 : 1, 2, 8(a)

9.1 : 3, 4, 6, 13

9.2 : 1, 3, 5

9.3 : 2, 3