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prime ideals

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a. Let R^+ be the set of positive real numbers. Define operations on this set by the a + b = ab and a x b = e^(ln (a)ln (b)), where the right hand sides have the usual meaning in the real numbers. Prove that R^+ is a field with these operations by showing that exp : R -----> R^+ is an isomorphism, where exp is the usual exponential function

b. Prove directly that every maximal ideal is prime.

c. For the ring of integers Z, describe all the prime ideals. Which ones are maximal, which ones are not?

d. For the ring Z_4 [x], describe all the prime ideals

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This solution fully describes prime ideals.