Discussion
Which of the following is false ?
Answer: A
The false statement is in a commutative ring with unity, every prime ideal is maximal.
In Z the zero ideal is prime (Z is an integral domain) but not maximal, since \((0) \subset (2) \subset \mathbf{Z}\).
The other three are true. Every maximal ideal M is prime because R/M is a field, hence an integral domain. In a Boolean ring, if P is prime then R/P is a Boolean integral domain; from \(x^{2} = x\), \(x(x - 1) = 0\), so every element is 0 or 1 and R/P is the field with two elements, making P maximal. Maximal implies prime in any commutative ring with unity, Boolean or not.
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