Consider a ring Z/2]={a+ b[/2] | a,bEZ}where addition and multiplication on ZI/21is defined as follows:(a + b[/2]) + (c + d[/Z]) = (a + c) + (b + d)[/Z](a + bl/2])(c+ d/2 ]) = (ac + 2bd) + (ad+ bc)[/2]Prove or disprove that(a) Z[2] is a commutative ring.(b) Z[/2] is a ring with unity.(c) Z[/2] is a field.

A ring is commutative if all the elements of the ring commute under the second operation. Clearly,…

Consider a ring ZI/2)={a + b[/2]|a,bEZ}where addition and multiplication on ZI/21is defined as (a + b[/2 ]) + (c + d/2]) = (a + c) + (b + d)[/2] (a + b[/2])(c+ d/7]) = (ac +2bd) + (ad+ bc)[/2] Prove or disprove that (a) Z[/2] is a commutative ring. (b) Z[/2] is a ring with unity. (c) Z[/2]is a field.

Consider a ring ZI/2)={a + b[/2]|a,bEZ}where addition and multiplication on ZI/21is defined as (a + b[/2 ]) + (c + d/2]) = (a + c) + (b + d)[/2] (a + b[/2])(c+ d/7]) = (ac +2bd) + (ad+ bc)[/2] Prove or disprove that (a) Z[/2] is a commutative ring. (b) Z[/2] is a ring with unity. (c) Z[/2]is a field.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.2: Integral Domains And Fields

Problem 2E: Consider the set ={[0],[2],[4],[6],[8]}10, with addition and multiplication as defined in 10. a. Is...

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