1. Suppose f: Z ® Z₆ is the function f(x) = [x]₆.

1.a. Show that f is a homomorphism.

1.b. Show that f is surjective.

1.c. Show that f is not injective.

2. Show that the rings Z₄ and Z₂ × Z₂ are not isomorphic.

3. Let D: R[x] ® R[x] be the derivative map given by D(a₀+a₁x+a₂x²+ ¼a_nxⁿ) = a₁+2a₂x+ ¼+na_nx^n-1. Show that D is not a homomorphism. (Hint: Compute D(x²).)

4. Suppose that R and S are rings, R' is a subring of R, and S' is a subring of S. Show that R' ×S' is a subring of R ×S.

5. Show that Z₆× Z₅ @ Z₁₀× Z₃.

6. Let R be a ring with identity. An element e Î R is called idempotent if e² = e. The elements 0 and 1 are called the trivial idempotents of R. All other idempotents (if any exist) are called nontrivial idempotents

Let R,S be rings with identity with R ¹ 0, S ¹ 0 (that is, neither R nor S is the ``stupid'' ring). Show that R×S always has nontrivial idempotents.

7. Find the solutions to the system of congruences

x º 3 (mod 5)

x º 1 (mod 6)

x º 2 (mod 11)

8. Let f: Z₈® Z₁₂ be given by f([x]₈) = [3x]₁₂.

(a): Show that f is a well-defined homomorphism of groups (under addition).

(b): Show (by example) that f is neither injective nor surjective.

(c): Is f a homomorphism of rings? Show why or why not.

9. Let G be an abelian group, with identity element e.

(a): Show that if a,b Î G, and aⁿ = e, b^m = e for some n,m Î N, then (ab)^nm = e.

(b): Show that H = {a Î G: a^k = e for some k ³ 1} is a subgroup of G.

10. Show that if G is a group, and H,K Í G are subgroups of G, then HÇK = {g Î G : g Î H and g Î K} is also a subgroup of G.