Mathematics 894 Problem Set 5

Questions due next week (Friday February 15).

1. Page 403 #1 (b) (c) Do diagram chasing as was done in class, giving also a written reason for each step.

2. Give explicitly two inequivalent extensions

[figure: 0; \frac{ℤ}{3ℤ}; f; \frac{ℤ}{9ℤ}; g; \frac{ℤ}{3ℤ}; 0]

State explicitly what the maps f, g are for your two sequences, and show that the two sequences are not equivalent. Are your sequences isomorphic?

3. Let R = ℤ[i\sqrt{5}] = \{a + bi\sqrt{5}|a, b ∈ ℤ\} and let I =< 1 + i\sqrt{5}, 2 > and J =< 1 − i\sqrt{5}, 2 > be ideals of R.

(a) Prove that R is a ring (starting from the description R = \{a + bi\sqrt{5}|a, b ∈ ℤ\}).

(b) Prove that I and J are not principal ideals (hence not free R-modules).

(c) Prove that IJ = 2R.

(d) Let π be the surjection of R-modules π : R ⊕ R → I given by π(1, 0) = 1 + i\sqrt{5}, π(0, 1) = 2. Define a homomorphism of R-modules g : I → R ⊕ R such that πg = 1_{I}. What can you conclude about I from these observations?

(e) Define and split a similar surjection R ⊕ R → J.

(f) (bonus) By combining the split surjections from parts (d), (e), or otherwise, prove that R ⊕ R ≅ I ⊕ J as R-modules.