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**curiousmuch** Let R be an integral domain where ideals are only of form aR={ar|r is an element of R} for some a in R. Prove that every nonzero prime ideal in R is a maximal ideal.

Besides knowing that every maximal ideal in a commutative ring with unity is a prime ideal, i do not know where to go.

How about this: assume that not every prime ideal in R is a maximal ideal. Then for some prime ideal I in R there is another prime ideal N such that I is contained in N and N is contained in R.

For some a,b in I then ab is in both I and N. This violates assumption that they are prime, so every prime must be maximal?

I have no idea if this is good and advice/help would be really great.