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?