# Integer ideals

**Algebra**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?