Show that Z module Q can't be written as direct sum of two non-zero submodule of Q.

Question

Show that the $\mathbb{Z}$-module $\mathbb{Q}$ cannot be written as direct sum of two non-zero submodules of $\mathbb{Q}$.

It's clear to me that intersection of any two submodules of $\mathbb{Q}$ has infinite intersection. But how to prove? Please help.

It would be helpful if you could edit your question to include the basic definitions that you are working with, and expand on the things that you have tried and what didn't work. — Xander Henderson, Dec 27 '17 at 02:44
Well, say $N, M$ are any two nonzero $\mathbb{Z}$-submodules of $\mathbb{Q}$. Pick $p/q \in N, r/s \in M$ both nonzero. Then $p \in N, r \in M$, and thus $N, M$ both contain the least common multiple of $p, r$. — Alex Wertheim, Dec 27 '17 at 02:48
Possible duplicate of Showing an abelian group indecomposable — , Dec 27 '17 at 13:14

score 8 · Accepted Answer · answered Dec 27 '17 at 02:50

Let's assume $\mathbb{Q} = M_1 \oplus M_2$, where $M_1$ and $M_2$ are nonzero submodules of $\mathbb{Q}$.

All we know about $M_1$ and $M_2$ are that they are nonzero, i.e., contain at least one nonzero element, so we start by taking $m_1 \in M_1$ and $m_2 \in M_2$ where $m_1$ and $m_2$ are both nonzero rationals.

We can write $m_1 = \frac{a_1}{b_1}$ and $m_2 = \frac{a_2}{b_2}$, where $a_1, a_2, b_1, b_2$ are nonzero integers.

Now we want to show that $M_1$ and $M_2$ have nontrivial intersection. Since both are $\mathbb{Z}$-submodules, we want to find some nonzero integer multiple of $m_1$ which is also a nonzero integer multiple of $m_2$. One that works is $a_1a_2 = a_2b_1m_1 = a_1b_2m_2$.

Show that Z module Q can't be written as direct sum of two non-zero submodule of Q.

1 Answers1