Question 4. the Cartesian product Z₂ Z5 of two ([k], [m]) and Given the triple < Z₂ XZ₁, , > defined in terms of sets of congruence classes, Z₂ and Z5, under operations ([1], [n]) = ([k+1], [m+n]) ([k], [m]) ([4], [n]) = ([kl], [mn]) (a) Prove that the first distributive law holds true. (b) Hence prove that is a ring. (c) Is it a commutative ring? Justify your answer.
Question 4. the Cartesian product Z₂ Z5 of two ([k], [m]) and Given the triple < Z₂ XZ₁, , > defined in terms of sets of congruence classes, Z₂ and Z5, under operations ([1], [n]) = ([k+1], [m+n]) ([k], [m]) ([4], [n]) = ([kl], [mn]) (a) Prove that the first distributive law holds true. (b) Hence prove that is a ring. (c) Is it a commutative ring? Justify your answer.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Question 4.
Given the triple < Z₂ x Z₁, B, D defined in terms of
the Cartesian product Z₂ Z5 of two sets of congruence classes, Z₂ and Zs, under operations
([k], [m])=([1], [n]) = ([k+l], [m+n])
and
([k], [m]) ([1], [n]) = ([kl], [mn])
(a) Prove that the first distributive law holds true.
(b) Hence prove that <Z₂ X Z₁, B, > is a ring.
(c) Is it a commutative ring? Justify your answer.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F114c4908-211d-434d-963e-fbb39e1c08fd%2Faae73f9a-969f-42de-9cd2-c40f25ead2b4%2Fjx4ypg_processed.png&w=3840&q=75)
Transcribed Image Text:Question 4.
Given the triple < Z₂ x Z₁, B, D defined in terms of
the Cartesian product Z₂ Z5 of two sets of congruence classes, Z₂ and Zs, under operations
([k], [m])=([1], [n]) = ([k+l], [m+n])
and
([k], [m]) ([1], [n]) = ([kl], [mn])
(a) Prove that the first distributive law holds true.
(b) Hence prove that <Z₂ X Z₁, B, > is a ring.
(c) Is it a commutative ring? Justify your answer.
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