4. Let Z[V2]= {a+b /2]a,b e Z}. Define addition and multiplication on Z[ /2] as follows: (a+b [/2]) + (c+d [2 ]) = (a+c) + (b+d)[ v2] (a+b [/2 ])(c+d [/2]) = (ac+2bd) + (ad+bc) /2 Prove or disprove the following statements: (a) Z[V2] is a ring. (b) Z[ /2] is a commutative ring. (c) Z[V2] is a ring with unity.
4. Let Z[V2]= {a+b /2]a,b e Z}. Define addition and multiplication on Z[ /2] as follows: (a+b [/2]) + (c+d [2 ]) = (a+c) + (b+d)[ v2] (a+b [/2 ])(c+d [/2]) = (ac+2bd) + (ad+bc) /2 Prove or disprove the following statements: (a) Z[V2] is a ring. (b) Z[ /2] is a commutative ring. (c) Z[V2] is a ring with unity.
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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![4. Let Z[V2]= {a +b/2]a,b e Z}. Define addition and multiplication on Z[V2] as
follows:
(a+b [V2 ]) + (c+d [ V2 ]) = (a+c) + (b+d)[ /2]
(a+b [VZ ])(c+d [/Z]
= (ac+2bd) + (ad+bc) /2
Prove or disprove the following statements:
(a) Z[ /2] is a ring.
(b) Z[V2] is a commutative ring.
(c) Z[V2] is a ring with unity.
(d) Z[V2] is a field.
(e) Z[ /2] is an integral domain.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1bc36662-e81e-4805-a9ba-e756ced76ed7%2Fe69dda4a-3ed0-4d1d-a57b-338d9135d284%2Fmd6tr6o_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4. Let Z[V2]= {a +b/2]a,b e Z}. Define addition and multiplication on Z[V2] as
follows:
(a+b [V2 ]) + (c+d [ V2 ]) = (a+c) + (b+d)[ /2]
(a+b [VZ ])(c+d [/Z]
= (ac+2bd) + (ad+bc) /2
Prove or disprove the following statements:
(a) Z[ /2] is a ring.
(b) Z[V2] is a commutative ring.
(c) Z[V2] is a ring with unity.
(d) Z[V2] is a field.
(e) Z[ /2] is an integral domain.
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