Math 464, Spring 2009

Elizabeth Dan-Cohen

Office hours: Tuesday 1:20-3:20, Thursday 1:20-2:20

Office: 456 Herman Brown

Phone: x5319

E-mail: edc@rice.edu

Course outline

• Ch I - p. 80 - problems 49, 50, 52

• Ch II - p. 115 - problems 2, 3, 6
• Let k and k' be fields. Show that every module over the ring k x k' is projective. Outline of a solution

• Ch III - p. 165 - problems 3, 4
• Problem on exact sequences of groups

• Ch III - p. 167 - problems 9, 10
• Ch XVI - p. 637 - problem 2

• Ch XVI - p. 613 - Prove Proposition 3.2(ii).
• - p. 625 - Prove Proposition 4.2.
• - p. 638 - problems 6, 7
• Give two proofs of the the fact that the tensor product of two projective modules is projective. One should use the fact that Hom(X@Y,Z) = Hom(X,Hom(Y,Z)), where @ should be the tensor product symbol.

• Ch XIX - p. 753 - problem 1
• Two problems

• Show that fiber coproducts exist in the category of A-modules. (Compare p. 81 problem 52.)
• Draw a diagram relating free, projective, flat, and injective Z-modules. In each region of your diagram, put an abelian group which has that set of properties.
• Let R be a ring. Prove that every R-module is projective if and only if every R-module is injective.
• Ch XX - p. 831 - problem 26

• Five problems

• Ch IX - p. 411 - problem 14 - Compare the same exercise from Eisenbud

• Three problems

• From handout on Grobner bases: 13, 19b, 32

• Five problems