HOMEWORK

Assignment 1

• Herstein. Page 152. 3,6
• Page 142. 5,6
• Page 166. 5
• Page 167. 8,9,13
• Let P be a prime ideal in the commutative ring R. Suppose I and J are two ideals in R such that IJ is contained in P. Show I or J (or both) is contained in P.
• Page 175. 5, 13

Assignment 2

• Herstein. Page 176. 16, 19, 21
• Page 183. 4,6,11,12,13,17
• Page 205. 10,12,14

Assignment 3

• Herstein. Page 190. 2,6(also infinite dimensional case),7-9
• Page 190. 10(also over F = Z/7),11
• Page 199. 2,4,5
• Given a finite complex of finite dimensional vector spaces over a field F show that the alternating sum of dimensions of the spaces V_i is equal to the alternating sums of the dimensions of the homology spaces.

Assignment 4 (I will give out the rest of this assignment on Monday)

• Herstein. Page 184. 17(infinite dimensional case).
• Page 200. 8,11(take V finite dimensional in 11).
• Page 205. 3,11,17.
• Suppose R is a ring with the following properties. It is a commutative ring with 1, AND every submodule of any free R-module is itself free. Show that R is a principal ideal domain. Hints: First of all, any ideal I in R is a submodule of R. Second, consider u,v in R both non-zero. Notice the stupid identity: uv + (-v)u = 0 which can be used to show that I is free of rank one.
• Three additional problems were distributed in class

Assignment 5

• Herstein, Page 268. 19, 20
• Page 290. 2,9,11,16
• Page 303, 5,7,12,18

Assignment 6

• Herstein, Page 303. 10,13,14,15,16,18,19
• Page 320. 2, 3, 17.

Assignment 7

• Herstein, Page 282. 5-9, 12, 16-18, 29
• Page 290. 8

Assignment 8

• Hand out with 7 computational problems
• Herstein. Page 215. 3,4,8,10,11

Assignment 9

• Herstein. Page 227. 5,6,10,12,15
• Page 236. 4,7,8,9,10,11
• Page 249. 1

Assignment 10

• Herstein. Page 249. 8,9,10
• other problems were distributed in class.