Explanation Consider the statement, “Only one maximal ideal exists for a given ring R .” Let R = ℤ 18 be a ring. The elements of ℤ 18 are as follows: ℤ 18 = { [ 0 ] , [ 1 ] , [ 2 ] , [ 3 ] , [ 4 ] , [ 5 ] , [ 6 ] , [ 7 ] , [ 8 ] , [ 9 ] , [ 10 ] , [ 11 ] , [ 12 ] , [ 13 ] , [ 14 ] , [ 15 ] , [ 16 ] , [ 17 ] } . Order of ℤ 18 is 18 . In the ring ℤ of integers, every ideal is the principal ideal, that is, I = ( n ) . The generators of ℤ 18 are the elements, which are relatively prime to 18 . The numbers, which are relatively prime to 18 , are 1 , 5 , 7 , 11 , 13 , 17 . Therefore, generators of ℤ 18 are [ 1 ] , [ 5 ] , [ 7 ] , [ 11 ] , [ 13 ] , [ 17 ] . By using the definition of maximal ideal, M is the maximal ideal of the commutative ring R , if M is not a proper subset of any ideal except R itself. Thus, ( [ 1 ] ) , ( [ 5 ] ) , ( [ 7 ] ) , ( [ 11 ] ) , ( [ 13 ] ) , ( [ 17 ] ) are not maximal ideals of ℤ 18 . Now, (i) ( [ 2 ] ) = { [ 2 ] , [ 4 ] , [ 6 ] , [ 8 ] , [ 10 ] , [ 12 ] , [ 14 ] , [ 16 ] , [ 0 ] } Order of ( [ 2 ] ) is 9

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ISBN: 9781285463230

Author: Gilbert, Linda, Jimmie

Publisher: Cengage Learning,

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Chapter 6.4, Problem 2TFE

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Whether the statement, “Only one maximal ideal exists for a given ring R ” is true or false.

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Whether the statement, “Only one maximal ideal exists for a given ring R ” is true or false.

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