# What does it mean for modules to be canonically identified?

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 $M\times N$

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Are you sure $M+N$ and $M\times N$ mean different things in this context? A $\mathbb Z$-module is nothing more nor less than an abelian group, except for the notation: modules use additive notation whereas (general) group theory tends to use multiplicative notation. And the "direct product of modules" is the natural generalization of "direct sum" of vector spaces. – Henning Makholm Sep 21 '11 at 13:48

I don't understand your notes, but the canonical identification of $M$ with a submodule of $M\times N$ is the embedding $M\to M\times N$, $m\mapsto (m,0)$, which is obviously injective and a module homomorphism.
EDIT: I guess your notes should be an identification $M\to M\times 0$
Since $M$ and $N$ are identified with submodules of $M\times N$ ($M$ is identified with the set of elements of the form $(m,0)$ for $m\in M$ and $N$ with the set of elements of the form $(0,n)$ with $n\in N$), you can form the sum $M+N$. This is the submodule of $M\times N$ consisting of sums $m+n$ with $m\in M$ and $n\in N$ (where you are identifying $m$ and $(m,0)$, and similarly $n$ with $(0,n)$). It's in fact clear that $M+N=M\times N$. In general, it only makes sense to talk about $M+N$ when both $M$ and $N$ are submodules of some larger module. In this case they are both submodules of $M\times N$.