MAT3702 Assignment 2 (COMPLETE ANSWERS) 2026 - DUE 19 August 2026

MAT3702 Assignment 2 (COMPLETE ANSWERS) 2026 - DUE 19 August 2026 ... 1. Prove that b | a if and only if (−b) | a. 2. If a | b and b | c, then prove that a | c. 3. Let R,S be rings and consider the following subsets of R × S R = {(r , 0S) | r ∈ R} and S = {(0R, s) | s ∈ S} where 0R ∈ R, 0S ∈ S are the zero elements in R,S respectively. (i) If R = ZZ3,S = ZZ5, then find R,S. (ii) For any rings R,S, show that R is a subring of R × S. (iii) For any rings R,S, show that S is a subring of R × S. 4. Prove that the intersection of a family of subrings of a ring R is a subring of R. 5. Let R be a ring and a ∈ R be a nonzero element which is not a zero divisor. Prove that (i) if ab = ac, then b = c. (ii) if ba = ca, then b = c. 6. Let f : R −→ S be a ring homomorphism and show that f (0R) = 0S, where 0R, 0S are the zero elements in R,S respectively. 7. Let R,S be rings and z : R −→ S be given by z(r ) = 0S for every r ∈ R and 0S ∈ S is the zero element in S. (i) Check as to whether z is a ring homomorphism or not (first check as to whether z is well-defined or not). (ii) If both R,S contain nonzero elements, then what can be said about z? 8. Let R,S be rings, where the operations on R × S are componentwise. (i) Prove that f : R × S −→ R given by f ((r , s)) = r is a surjective homomorphism (first check if f is well-defined or not). (ii) Prove that g : R × S −→ S given by g((r , s)) = s is a surjective homomorphism (first check if f is well-defined or not). (iii) If both R,S are nonzero rings, then prove that the homomorphisms f , g are not injective. 9. Let R be a ring with identity. If u ∈ R is a unit, then prove that u ∈ R is not a zero dovisor.