Been trying to figure this out for a while now (at least a week). The Problem Set was already handed in but I’m still trying to figure this out since I wasn’t able to answer this prior to handing it in. I really want to understand this better. Here’s the problem.
Let $0\to A\to B\to C\to 0$ be a short exact sequence of Modules. If $M$ is any module, prove that there are exact sequences.
$0\to A\oplus M\to B\oplus M\to C\to 0$
$0\to A\to B\oplus M\to C\oplus M\to 0$.