9 Questions - Introduction to Abstract Algebra - Final Examination | MATH 444, Exams of Abstract Algebra

Download 9 Questions - Introduction to Abstract Algebra - Final Examination | MATH 444 and more Exams Abstract Algebra in PDF only on Docsity! Newberger Math 444 Fall 2001 Final Exam Non-Cumulative Part name: 1. a. (8 pts) Define a homomorphism of groups. b. (8 pts) Define a homomorphism of rings. c. (10 pts) Let f : Z6 → Z24 be given by f([x]6) = [8x]24. Prove f is a homomorphism of groups (under +) but not a homomor- phism of rings. d. (10 pts) Let f : Z8 → Z24 be given by f([x]8) = [9x]24. Prove g is a homomorphism of rings. 2. (a) (8 pts) Define the kernel of a homomorphism of groups. (b) (16 pts) Prove Proposition 3.3.4, which says: Let G1 and G2 be groups, and let φ : G1 → G2 be a homomorphism of groups. Then φ is one-to-one if and only if ker(φ) = {e}. (c) (8 pts) Let φ : S3 → Z6 be a homomorphism of groups. Prove that ker(φ) 6= {(1)}, where (1) is the identity in S3. 3. Let R = { n 2m : n,m ∈ Z } . (a) (10 pts) Show that (R, +, ·) is a commutative ring with 1. (b) (6 pts) Is R a field? Explain. (c) (6 pts) Is R an integral domain? Explain. 4. A subset I ⊆ R of a ring R is called closed under scalar multipli- cation if for all r ∈ R and all i ∈ I, we have ri ∈ I. (a) (5 pts) Let I = 2Z and R = Z. Prove I is closed under scalar multiplication. (b) (5 pts) Let R and S be rings and φ : R → S be a homomor- phism of rings. Prove ker(φ) is closed under scalar multiplica- tion. Final Exam Cumulative Part 1. Let G = {(x, y) : x, y ∈ R with x 6= 0}, and consider the binary operation ∗ given by (x, y) ∗ (z, w) = (xz, xw + y) for (x, y), (z, w) ∈ G. (a) (10 pts) Prove that (G, ∗) is a group. (b) (10 pts) Prove that H = {(1, y) : y ∈ R} is a subgroup of G (remember to use the operation *). 2. (a) (8 pts) Prove Z3 × Z2 is cyclic. (b) (8 pts) Prove Z2 × Z2 is not cyclic. 1