Homework 2 for MATH-UA.0343-1: Algebra 1 - Spring 2012

Write up and turn in solutions to the highlighted problems from the list below. However, solve all other problems, as well.

The quizzes will cover the complete list of homework problems.

Due date: Wednesday, February 8, in the mailbox of Lukas Koehler. NO LATE HOMEWORK WILL BE ACCEPTED.

Problems: (page numbers refer to the textbook Herstein - Topics in Algebra - 2nd Edition, Wiley)

1. Let (R,+,·) be a commutative ring with identity. Show that (R,+) is a group but (R,·) is never a group. The solution to this problem should be turned in.
2. Fix an integer n>1.
   • Show that [x]·[y]=[x· y] is a well defined binary operation on Z_n.
   • An equivalence class [x] in Z_n is called a unit if there exists an equivalence class [y] in Z_n such that [x]·[y]=[1]. Show that [x] is a unit if and only if x and n are relatively prime.
   • Let U_n be the set of all units in Z_n. Show that (U_n,·) is an Abelian group.
   • Compute U₁₂.
3. Pages 35-37 , problems 2, 3, 6, 7, 8,10, 11, 22, 26.