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.
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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![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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8a9bbc7c-a95e-4b48-ac30-25422732479a%2Fa7509e2e-8b19-46a4-9af8-dca531eee75f%2Fdi5t22v_processed.png&w=3840&q=75)
Transcribed Image Text: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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