In GTM4, the author asked the reader to give a procedure for calculating the maximal essential extension of $A$ in $B$, where $B$ is a finitely generated abelian group.
I've got completely no idea of how to do so. Even for $B=\mathbb Z^n$, I can not find a general procedure to compute maximal essential extension.
I am also working on GTM4. To this exercise, I've come up with a coarse solution, and am still looking for potential better solutions:
By the structure theorem of f.g. modules on PID's, $B$ has the form $ T (B) \oplus F (B)$. For any subgroup $A \subset B$, consider the projection $A \rightarrow F (B)$. There's a well-known theorem on submodules of free modules over a PID:
For any free module $M$ over $A$ (PID) and any submodule $N \subset M$, there is a representation of $M$ by a direct sum of $A$,
$M = \oplus_{i \in I} A_i$, where $A_i = A$ and $N = \bigoplus_{i \in I} c_i A_i$ with the obvious inclusion, where $c_i \in A$. ($c_i$ could be zero, corresponding to the case where $F (M / N) \neq 0$).
Denote $\mathrm{im} (A \rightarrow F (B)) =: C$. By the aforementioned theorem, we can express $F (B) = \bigoplus_{i \in I_1} \mathbb{Z} \oplus \bigoplus_{i \in I_2} \mathbb{Z}$, with $C = \bigoplus_{i \in I_1} c_i \mathbb{Z} \quad (c_i > 0)$, embedding in $F (B)$ via the obvious map sending the $i$-th generator in $C$ to $c_i$ times the $i$-th generator in $F (B)$. Then $B = \left( T (B) \oplus \bigoplus_{i \in I_1} \mathbb{Z} \right) \oplus \left( \bigoplus_{i \in I_2} \mathbb{Z} \right)$ Then by exercise 9.2, the maximal essential extension of $A$ lies in $T (B) \oplus \bigoplus_{i \in I_1} \mathbb{Z}$. For simplicity of notation we may assume $I_2 = \varnothing$, and just rename $I_1$ to $I$.
I claim a easy to prove criterion for deciding whether potential $D, A \subset D \subset B$ can be an essential extension. For simplicity let's assume $T (A)$ is already maximal by potentially extending $A$ by any maximal essential extension in $T (B)$. Then $D$ can be an essential extension of $A$ iff $D \cap T (B) = T (A)$: use the condition $I_2 = \varnothing$.
Given any $D \quad (A \subset D \subset B)$, if we want to check whether $D$ could be enlarged, we only need to check elements $t + \sum_{i \in I} a_i x_i$ where $x_i \in B$ projects to the $i$-th generator in $F (B)$ and $t \in T (B)$ with the properly $0 \leqslant a_i < c_i$, since every element in $B$ can be written as an element of such form plus an element in $A$. There are only finitely many such elements. So we may try to add them to $A$ one by one and see if it violates the criterion above. Checking this involves solving some linear diophantine equations, which can be done mechanically.
This is a really dumb and inefficient approach, but at least it guarentees to find a maximal essential extension in finite time. I'm also looking forward to better approaches.