Question
Suppose that R is a unique factorization domain and that a,b ?R are distinct irreducible elements. Which of the following statements is TRUE

The ideal 〈1+a〉 is a prime ideal

The ideal 〈a+b〉 is a prime ideal

The ideal 〈1+ab〉 is a prime ideal

The ideal 〈a〉 is not necessarily a maximal ideal
Answer is
"The ideal <a> is not necessarily a maximal ideal"
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