Explanation Formula used: 1) Theorem: Let R be a commutative ring with unity, and let M be an ideal of R . Then R / M is a field if and only if M is a maximal ideal of R . 2) Every field is an integral domain. 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 J be an ideal of E such that ( 14 ) ⊂ J ⊆ E . Let x be any element of J that is not in ( 14 ) , such that x = 14 k + 2 = 2 ( 7 k + 1 ) where k ∈ ℤ

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Chapter 6.4, Problem 20E

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To prove: The ideal (14) is a prime ideal of E.

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To prove: The ideal ( 14 ) is a prime ideal of E .

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