In Attiyah commutative algebra page 71, it is given some equivalent definitions of Jacobson ring. One of the definitions are that every prime ideal which is not maximal is equal to the intersection of prime ideals which contains it strictly. How is this compatible with the prime avoidance lemma? I.e if a prime ideal is equal to the intersection of a family of ideals then one of the ideals is the given prime ideal.
is every prime ideal in a Jacobson ring maximal?