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Question 6. (a) Give an example of a commutative ring without zero divisors that is not an integral domain. If there is no such example, then state so. (b) Give an example of a field that is not an integral domain. If there is no such example, then state so. (c) Prove or disprove: The ring Z. , > is isomorphic to Z₂ × Z₁, 0, 0. A, +, with underlying set A = {m+n√-2: m,n € Z}, and a with underlying set B = {m+n√-3: m,n €Z}. Given a ring A = ring B =

Question 6. (a) Give an example of a commutative ring without zero divisors that is not an integral domain. If there is no such example, then state so. (b) Give an example of a field that is not an integral domain. If there is no such example, then state so. (c) Prove or disprove: The ring Z. , > is isomorphic to Z₂ × Z₁, 0, 0. A, +, with underlying set A = {m+n√-2: m,n € Z}, and a with underlying set B = {m+n√-3: m,n €Z}. Given a ring A = ring B =

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter8: Polynomials
Section8.1: Polynomials Over A Ring
Problem 17E
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Question 6.5 (a) Give an example of a commutative ring without zero divisors that is not an integral domain. If there is no such example, then state so. (b) Give an example of a field that is not an integral domain. If there is no such example, then state so. (c) Prove or disprove: The ring Z, , > is isomorphic to Z₂ × Z₂, 0, 0. A, +, with underlying set A = {m+n√2: m,n € Z}, and a with underlying set B = {m+n√-3: m, n € Z}. Given a ring A = ring B =<B,+, (d) Show that the mapping 6: A → B defined by 0(m+n√-2)=m+n√-3 is one-to-one and onto. Show that the mapping is not a ring isomorphism. 2
Transcribed Image Text:Question 6.5 (a) Give an example of a commutative ring without zero divisors that is not an integral domain. If there is no such example, then state so. (b) Give an example of a field that is not an integral domain. If there is no such example, then state so. (c) Prove or disprove: The ring Z, , > is isomorphic to Z₂ × Z₂, 0, 0. A, +, with underlying set A = {m+n√2: m,n € Z}, and a with underlying set B = {m+n√-3: m, n € Z}. Given a ring A = ring B =<B,+, (d) Show that the mapping 6: A → B defined by 0(m+n√-2)=m+n√-3 is one-to-one and onto. Show that the mapping is not a ring isomorphism. 2
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