Integer idealsAlgebra Level 5
A nonempty set \( I \) in a ring \( R \) is called an ideal if
(1) it is closed under addition: \( a\in I, b\in I \Rightarrow a+b \in I \)
(2) it "swallows up" under multiplication: \( a \in R, i \in I \Rightarrow ai \in I \).
A proper ideal is one that is not equal to the entire ring.
A proper ideal is prime if \( ab \in I \) implies that \( a\in I \) or \( b \in I \).
A proper ideal is maximal if there are no ideals in between it and the entire ring: if \( J \) is an ideal, then \( I \subseteq J \subseteq R \) implies \( I=J\) or \( J=R\).
How many ideals of \( \mathbb Z \) are prime but not maximal?