Explanation Given information: ( n ) is an ideal of ℤ such that n ≠ 1 , ( n ) ≠ { 0 } . Formula used: 1) Definition of prime ideal: An ideal I of a commutative ring R is a prime ideal if I ≠ R and if a b ∈ I implies either a ∈ I or b ∈ I . 2) Theorem: For n > 1 , ℤ n is an integral domain if and only if n is a prime. 3) If I is an ideal of a commutative ring R with unity such that I ≠ R , I ≠ { 0 } then R / I is an integral domain if and only if I is a prime ideal. Proof: Let n be a prime integer and I = ( n ) such that n ≠ 1 , ( n ) ≠ { 0 } . As ℤ is commutative ring with unity and I = ( n ) is an ideal of ℤ , By using theorem, for n > 1 , ℤ n is an integral domain if and only if n is prime

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Author: Gilbert, Linda, Jimmie

Publisher: Cengage Learning,

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Chapter 6.4, Problem 14E

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To prove: For n≠1, (n)≠{0}, an ideal (n) of ℤ is a prime ideal if and only if n is a prime integer.

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Chapter 6.4, Problem 13E

Chapter 6.4, Problem 15E

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To prove: For n ≠ 1 , ( n ) ≠ { 0 } , an ideal ( n ) of ℤ is a prime ideal if and only if n is a prime integer.

Solution Summary: The author proves that an ideal (n) is a prime ideal if and only when

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To prove: For n≠1, (n)≠{0}, an ideal (n) of ℤ is a prime ideal if and only if n is a prime integer.

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Chapter 6 Solutions