I am not sure I am understanding a defintion and would like some input on if I am using it correctly.
Let $U_p$ be the sub-module of the $\mathbb{Z}$-module $\mathbb{Q}/\mathbb{Z}$ consisting of the classes mod $\mathbb{Z}$ of rational numbers of the form $k/p^n$ with $k \in \mathbb{Z} , n\in \mathbb{N}$ for some fixed prime p.
Let $E$ be the product $\mathbb{Z}$-module $M \times N$ where $M$ and $N$ are isomorphic to $U_p$ and $M$ and $N$ are canonically identified with sub-modules of $E$.
What does it mean for $M$ and $N$ are canonically identified with sub-modules of $E$.
I have in my notes there is a canonical identification $j: M \rightarrow M+N$ but I am confused because $M+N$ is not in the set $M\times N$