MATH21112 Rings and Fields
Example Sheet 5
Homomorphisms and Isomorphisms
1. Prove that isomorphism is an equivalence relation on the set of all rings.
2. Show that the ring of 2 × 2 matrices of the form.
with a,b,c, d ∈ R (and i a square root of —1) is isomorphic to the ring of quaternions (see Example 2.15).
3. Show that all rings with three elements are isomorphic. (Hint: consider the + and × tables.)
4. Show that the projection map π 1 : R1 ×R2 :—→ R1 given by π1 ((r1 , r2 )) = r1 is a homomorphism. Is it an isomorphism?
5. Let θ : Znk —→ Zn be defined by θ([a]nk ) = [a]n , where n, k ∈ N, n ≥ 2. Prove that θ is a homomorphism. Prove that there is no homomorphism from Zm to Zn if n is not an integer factor of m.
(Hint: consider the characteristics of the rings.)
6. Show that if θ : R —→ S and β : S —→ T are homomorphisms such that the composition β o θ : R —→ T is an embedding then it need not be the case that β is an embedding (see Lemma 3.10).