Mathematics 811 Assignment 1

General Instructions Do not hand in the A problems. These will be “review” or easy questions. But look them over and do them if they seem new to you. Ask me about any difficulties. Each B problem will be assigned to one person to hand in and to prepare solutions for our web page. (Your solution can either be TeXed already, or handwritten and then TeXed after my comments.) Do the rest of the B problems as you see fit and be sure to read the solutions. Some results from the B problems will be used later in the course. Hand in all the C problems. One person will later be asked to prepare the solution for posting. The Challenge Problems are either more difficult, or I am not sure of the answer. Challenge Problem. Show that the ring

( \begin{matrix}ℤ & ℚ \\ 0 & ℚ \end{matrix} ) = \{( \begin{matrix}a & λ \\ 0 & μ \end{matrix} ) | a ∈ ℤ, λ, μ ∈ ℚ\}

is right Noetherian but not left Noetherian. bf B1(a): Exercise 1A p3 of Tony’s notes (Adam) B1(b): Find a free resolution for ℤ/2ℤ as a ℤ/8ℤ-module. (Adam) A1: Exercise 1B p5 Tony’s notes. A2: Exercise 1C p6 Tony’s notes. A3: Exercise 2A p8 Tony’s notes. A4: Exercise 2B p8 Tony’s notes. A5: Exercise 2C p12 Tony’s notes. (show that I is not principal). A6: Exercise 2D Tony p11. B2: Exercise 3A Tony p16 (Brian) B3: Exercise 4A p19 (David) B4: Exercise 4C Tony p20 (Susan) B5: Exercise 4D Tony p22 (Laura parts 1 and 2, Sindi parts 3 and 4) B6: Exercise 6A Tony p30. Furthermore show that in any category the 0 object is unique up to isomorphism. (David) B7: Exercise 6B Tony p30. (Ping)