**1.** Suppose f: Z ® Z_{6} 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 Z_{4} and
Z_{2} × Z_{2}
are not isomorphic.

**3.** Let D: R[x] ® R[x] be the
derivative map
given by D(a_{0}+a_{1}x+a_{2}x^{2}+ ¼a_{n}x^{n}) = a_{1}+2a_{2}x+ ¼+na_{n}x^{n-1}.
Show that D is not a homomorphism. (Hint: Compute D(x^{2}).)

**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 Z_{6}× Z_{5} @ Z_{10}× Z_{3}.

**6.** Let R be a ring with identity. An element e Î R
is called *idempotent *if e^{2} = 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: Z_{8}® Z_{12} 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 a^{n} = 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.

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