# Can we deduce that $M_0$ is a submodule of the limit of the following diagram?

Let $$M_0$$ be an R-module, and suppose $$M_{n+1}$$ is the pushout of the diagram below as shown, for all $$n \in \mathbb{N}$$:

$$\begin{array}{ccc}M_n&\to& M_{n+1}\\\uparrow &&\uparrow\\A&\to& B\end{array}$$

Define $$M:= colim(M_0 \rightarrow M_1 \rightarrow M_2 \rightarrow.....)$$

Can we deduce that $$M_0$$ is a submodule of $$M$$? What about if the left arrow is surjective and the bottom arrow is injective? Would this help?

I am asking this in order to complete my understanding of the proof that $$R-mod$$ has enough injectives. In this proof, A and B are given, however there is no justification given for the claim which I am inquiring about, it is just stated to be obvious. Thanks.

Then, if $$A\to B$$ is injective, then $$M_i$$ embeds to $$M_{i+1}$$, and it's easy to see that $$M':=\bigcup_iM_i$$ satisfies the universal property of the colimit, hence $$M\cong M'$$ and the natural arrow $$M_i\to M$$ is injective for all $$i$$.