Mathematics 894 Problem Set 4

Suggested reading: Sections 10.4, 10.5. You might also review sections 10.1-10.3 on elementary module theory. The questions in this problem set do not require R to be commutative. However you may assume that R is commutative if you feel more comfortable with this assumption.

Questions due next week (Friday February 8).

1. Page 403: #3, 4.

2. Page 403: #6.

3. Page 404: #9(b), 11(c). You may use previous parts of these questions.

4. Page 404 #14(b). (Note that for the “only if” part, no assumption has been made about the original sequence on page 404, so you will have to prove that ψ is injective, that φ is surjective, and that im(ψ)=ker(φ), as well as proving the splitting). Hint: try various choices of D such as D = L, D = N/im(φ) etc.