Exercise 11, § 3

Let A and B be two infinite sets of rational prime numbers, with no common elements, and not containing the number 5. Let $(e_i)_{1 \leq i \leq 4}$ be the canonical basis of the $\mathbf{Q}$-vector space $\mathbf{Q}^4$. a) Let M be the Z-submodule of $\mathbf{Q}^4$ generated by the elements $e_1/m$ for $m \in A$; let N be the Z-submodule of $\mathbf{Q}^4$ generated by the elements $e_2/m$ with $m \in A$, $e_3/n$ with $n \in B$, and $(e_1 + e_3)/5$. Then $M \cap N = 0$. Put $e'_1 = 8e_1 + 3e_2$ and $e'_2 = 5e_1 + 2e_2$. Let $M'$ be the Z-submodule of $\mathbf{Q}^4$ generated by the elements $e'_1/m$ with $m \in A$; let $N'$ be the Z-submodule of $\mathbf{Q}^4$ generated by the elements $e'_2/m$ with $m \in A$, $e_4/n$ with $n \in B$, and $(3e'_2 + e_3)/5$. Show that $M' \cap N' = 0$, that $M \oplus N = M' \oplus N'$; that M and $M'$ are isomorphic and indecomposable; and that N and N' are indecomposable but not isomorphic (compare VII, p. 69, Ex. 23). b) Let M be the Z-submodule of Q^4 generated by the elements e_1/m with m \in A, e_2/n with n \in B, and (e_1 + e_2)/5; let N be the Z-submodule of Q^4 generated by the elements e_3/m with m \in A, e_4/n with n \in B, and (e_1 + e_4)/5. Put e'_1 = 2e_1 + e_3, e'_2 = e_2 + 3e_4, e'_3 = 17e_1 + 9e_3, and e'_4 = e_2 + 2e_4. Let M' be the Z-submodule of Q^4 generated by the elements e'_1/m for m \in A, e'_2/n for n \in B, and (e'_1 + 2e'_2)/5; let N' be the Z-submodule of Q^4 generated by the elements e'_3/m for m \in A, e'_4/n for n \in B, and (e'_3 + 2e'_4)/5. Show that M \oplus N = M' \oplus N', that M is isomorphic to N and M' is isomorphic to N', but M' is not isomorphic to M. c) Let M be the Z-submodule of Q^4 generated by e_1/5^n (n \geq 0); let N be the submodule of Q^4 generated by e_2/5^n, e_3/2^n, e_4/3^n (n \geq 0), (e_2 + e_3)/3 and (e_2 + e_4)/2. Put e'_1 = 3e_1 - e_2 and e'_2 = 2e_1 - e_. Let M' be the Z-submodule of Q^4 generated by the elements e'_1/5^n, e_3/2^n (n \geq 0) and (e'_1 - e_3)/8; let N' be the Z-submodule of Q^4 generated by the elements e'_2/5^n, e_4/3^n (n \geq 0) and (e'_2 - e_4)/2. Show that M \oplus N = M' \oplus N', that M and N' are indecomposable of rank 2, and that M has rank 1.