Given the set of integers mod m denoted Z, the elements of Za are denoted [x]m, where x is aninteger from 0 to m – 1. Each element [x], is an equivalence class of integers that has the sameinteger remainder as x when divided by m.Consider, for example, Z; = {[0], [1], [2], [3], [4];, [5];, [6];}. The element [5], represents theinfinite set of integers of the form 5 plus an integer multiple of 7. That is,[5],= {. . . -9, -2, 5, 12, 19, 26, ..},or, more formally,[5], = {y. y = 5 + 7q for some integer q}.Modular addition, +, is defined on the set Z in terms of integer addition as follows: []. + [b], =[a + blm.Modular multiplication, *, is defined on the set Za in terms of integer multiplication as follows: [a].[b], = [a * bl..The set Z with these operations is a ring for any integer m.1. Choose a two-digit even integer for m and, using a counterexample, explain why Za with theoperations of + and * is not an integral domain.

We need to prove that ℤm is a ring for any integer m but not a integral domain. Definition of ring:…

Answered: 1. Choose a two-digit even integer for…

1. Choose a two-digit even integer for m and, using a counterexample, explain why Za with the operations of + and * is not an integral domain.

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

Given the set of integers mod m denoted Z, the elements of Za are denoted [x]m, where x is an
integer from 0 to m – 1. Each element [x], is an equivalence class of integers that has the same
integer remainder as x when divided by m.
Consider, for example, Z; = {[0], [1], [2], [3], [4];, [5];, [6];}. The element [5], represents the
infinite set of integers of the form 5 plus an integer multiple of 7. That is,
[5],= {. . . -9, -2, 5, 12, 19, 26, ..},
or, more formally,
[5], = {y. y = 5 + 7q for some integer q}.
Modular addition, +, is defined on the set Z in terms of integer addition as follows: []. + [b], =
[a + blm.
Modular multiplication, *, is defined on the set Za in terms of integer multiplication as follows: [a].
[b], = [a * bl..
The set Z with these operations is a ring for any integer m.
1. Choose a two-digit even integer for m and, using a counterexample, explain why Za with the
operations of + and * is not an integral domain.

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