Math 310 Some practice problems for Exam 2

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+ ¼anxn) = a1+2a2x+ ¼+nanxn-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, bm = e for some n,m Î N, then (ab)nm = e.

(b): Show that H = {a Î G: ak = 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.

File translated from TEX by TTH, version 0.9.