M953 Homework 5 Solutions

Due Friday, March 8, 2002

[1] Let I and J be ideals in a ring R. Let f : R -> R/I x R/J be the homomorphism f(r) = ([r]I,[r]J), where, for example, [r]I means the equivalence class r + I in R/I. Denote the intersection of I with J by Int(I,J). Note: What we have actually done here is to show that there is a short exact sequence 0 -> R/Int(I,J) -> R/I x R/J -> R/(I+J) -> 0 of additive groups. [A sequence of maps is exact if the image of each map is the kernel of the next one; it is called short if it has three terms in the middle with zeros on the end.]

Solution: [2] Solution:

[3] In this problem you may assume that k is algebraically closed, if you like. Solution:

Note: To gather some data for [3](f), use Macaulay2. Here are some commands that may help. First make sure you've defined your ring; say, R = ZZ/3[x,y] or R = ZZ/31991[x,y,z].