Homework for Math511B

01/16:
(Do not hand in this one)  Chapter 4. Ex. 1.4, read sections 1 and 2 (up to the definition of free modules) carefully. Pay special attention to the definition of direct sum, direct product and free modules (the book defines it using universal properties). If this is the first time you see all these, work out the proofs of the theorems that are omitted from the book.

01/21: Chapter 2, Ex. 7.3, 8.39. Ex: Let R be a ring and M be a Noetherian module over R. If M ---> M is a surjective R-homomorphism, then it is an isomorphism.

01/28:Chapter 4, prove Theorem 2.3. Exercise 7.2, 7.6.

01/30: Chapter 4, Exercise 3.2, 7.10, 7.12.

02/06: Chapter 4, prove Proposition 3.5 (ii). Exercise 7.15, 7.16.

02/13: Chapter 4, Exercise 7.29, 7.30 (x)(xi)(xii), 7.35 (iii)(iv)

02/18: Chapter 4, Exercise 6.3, 6.4, 6.5

02/25: Finish the proof of the snake lemma, the nine lemma and the five lemma. I forgot to assign homework last Thursday, but the qualifying exam problem Aug 2019 5B is recommended.

02/27: Work out the exactness of Hom in full detail.

03/05: Work out the HW from the class. Also Chapter 5, Exercise 5.7, 5.8.

03/24: Show that the two definitions of the (left) modular ideals are the same. Re-do the problems from the midterm exam.

03/26: Chapter 5, Exercise 1.4, 2.1, 2.2, 5.1, 5.2

03/31: Chapter 5, Exercise 2.3, 5.3, 5.4, 5.5.

04/02: Chapter 5, Exercise 3.1, 5.6, 5.10.

04/07: Chapter 5, Exercise 4.1, 5.17, (optional 5.19, 5.20, 5.21).

04/09: Find all simple modules over M_n(D), D a division ring.

04/14: (1) If I is a minimal ideal of R and I^2 = I, show that I is a simple ring. (2) Find all semisimple rings with 324 elements. (optional: do all qualifying exam problems related to semisimple artinian rings)