Question 1. State whether True or False. Provide a reason in each case. (a) The pair (Z321.-) that consists of the set of congruence classes modulo 321 together with multiplication, has two zero divisors. (b) Every integral domain consists of an Abelian additive group, and a commutative unital ring which has a unity, but no zero divisors. (c) The set nZ for n € N, the set R\ Q. and the set C\R have no zero divisors.
Question 1. State whether True or False. Provide a reason in each case. (a) The pair (Z321.-) that consists of the set of congruence classes modulo 321 together with multiplication, has two zero divisors. (b) Every integral domain consists of an Abelian additive group, and a commutative unital ring which has a unity, but no zero divisors. (c) The set nZ for n € N, the set R\ Q. and the set C\R have no zero divisors.
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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![Question 1.
State whether True or False. Provide a reason in each case.
(a) The pair (Z321,-) that consists of the set of congruence classes modulo 321 together with
multiplication, has two zero divisors.
(b) Every integral domain consists of an Abelian additive group, and a commutative unital
ring which has a unity, but no zero divisors.
(c) The set nz for n N, the set R\Q, and the set C\R have no zero divisors.
(d) Given the set Z24 of congruence classes modulo 24. Then the congruence classes [2], [3], [4],
[6], [9], [12], [18], and [20] are some of the zeros divisors of Z24.
(e) The triple (Z[√-3], +,-) consisting of the set of algebraic numbers
Z[√-3] = {+ + √-3: for k, l, m, n € Z and I, n ‡ 0}
with operations of addition + and multiplication, constitutes an integral domain.
(f) The set of all functions M ([0,1]) from the the closed unit interval [0, 1] onto [0, 1],
together with usual function addition and function multiplication, does not have any zero
divisors. If you think it does, then give an example.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb685f800-bd36-484b-82e6-b7cb9e75277f%2Fcf3d7b58-106b-4155-a26c-d22c79f0af63%2Fndtyp6_processed.png&w=3840&q=75)
Transcribed Image Text:Question 1.
State whether True or False. Provide a reason in each case.
(a) The pair (Z321,-) that consists of the set of congruence classes modulo 321 together with
multiplication, has two zero divisors.
(b) Every integral domain consists of an Abelian additive group, and a commutative unital
ring which has a unity, but no zero divisors.
(c) The set nz for n N, the set R\Q, and the set C\R have no zero divisors.
(d) Given the set Z24 of congruence classes modulo 24. Then the congruence classes [2], [3], [4],
[6], [9], [12], [18], and [20] are some of the zeros divisors of Z24.
(e) The triple (Z[√-3], +,-) consisting of the set of algebraic numbers
Z[√-3] = {+ + √-3: for k, l, m, n € Z and I, n ‡ 0}
with operations of addition + and multiplication, constitutes an integral domain.
(f) The set of all functions M ([0,1]) from the the closed unit interval [0, 1] onto [0, 1],
together with usual function addition and function multiplication, does not have any zero
divisors. If you think it does, then give an example.
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