Just as the title says im trying to prove that the infinite direct product of integers with itself is not an essential Z-submodule of the infinite direct product of the rational numbers with itself. Essential meaning that its intersection with any other nonzero submodule is nonzero. What this basically comes down to is finding a submodule that does not contain any nonzero integer sequences. Im asking for any advice towards this direction.
2026-03-25 06:06:51.1774418811
infinite direct product of integers is not an essential submodule of infinite direct product of rationals.
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Hint: If a submodule $N$ with the property you are looking for exists, then there exists such a submodule $N$ which is cyclic (just take the submodule generated by any nonzero element of $N$). So you may as well look for an example of $N$ that is cyclic. That is, you want to find some element of the module such that no nonzero multiple of it has all integer coordinates.