I'm trying to prove a rather ambiguous statement:
"Prove that the direct sum of any collection of torsion modules is a torsion module."
This is what I have so far:
"Let $\{M_i\}_{i \in I}$ be a collection of torsion modules.
Additionally, let $m_i \in M_i$ for each $i \in I$ be torsion elements.
Then we have that there exist nonzero $r_i \in R$ such that $r_im_i=0_M$ for each $i \in I$.
Consider $\oplus_{i \in I}M_i$…"
I'm unsure of where to go from here. Any tips?
Best Answer
Hint: The general element of $\bigoplus_{i\in I}M_i$ "looks like" a tuple $(a_i)_{i\in I}$, with $a_i\in M_i$ for each $i$, and $a_i=0$ for all but finitely many $i$. So you can find finitely many $i_1,\ldots,i_n\in I$ such that if $j\notin \{i_1,\ldots,i_n\}$, then $a_j=0$ already. That means you only have to worry about making sure you can annihilate $a_{i_1}$, $a_{i_2},\ldots, a_{i_n}$. Can you do that?