Question 7. (a) The mapping 8: C→C defined by 0(2) iz for i² = -1 and z EC Is a homomorphism? Determine the kernel if it is a homomorphism. (b) An ideal M of a ring R = R, +, > is a maximal ideal if MR and there is no ideal I of R such that MGIR. Prove that if R is a commutative ring with unity e, and I is an ideal of R, then I is a maximal ideal if and only if R/T is a field. (c) Show that the Euclidean domain E, +, is a unique factorisation domain. (d) Express the polynomial 3r³2r² + 3x - 2 as a product of factors over R. Then express this polynomial as a product of factors over C.

Question 7. (a) The mapping 8: C→C defined by 0(2) iz for i² = -1 and z EC Is a homomorphism? Determine the kernel if it is a homomorphism. (b) An ideal M of a ring R = R, +, > is a maximal ideal if MR and there is no ideal I of R such that MGIR. Prove that if R is a commutative ring with unity e, and I is an ideal of R, then I is a maximal ideal if and only if R/T is a field. (c) Show that the Euclidean domain E, +, is a unique factorisation domain. (d) Express the polynomial 3r³2r² + 3x - 2 as a product of factors over R. Then express this polynomial as a product of factors over C.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter6: More On Rings

Section6.4: Maximal Ideals (optional)

Problem 13E

See similar textbooks