# 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?

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