An Ascending Chain Of Ideals In The Integers

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Problem 30 (An Ascending Chain Of Ideals In The Integers)

Let $R=\mathbb{Z}$. Consider the ideal $A_1=\left<56\right>.$

1. Find an ascending chain of ideals consisting of three ideals $A_1\subsetneq A_2\subsetneq A_3$ with $A_3=R$.
2. What is the largest $n$ such that $A_1\subsetneq A_2\subsetneq A_3\subsetneq \cdots\subsetneq A_n=R$ is an ascending chain of ideals? How many such chains are there of this length?
3. If $n\in \mathbb{Z}$, describe a process we could use to find a longest possible ascending chain of ideals with $\left<n\right>\subsetneq A_2\subsetneq \cdots\subsetneq A_n=\mathbb{Z}$.

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