Understanding the definition of direct limit

Question

The attached snapshot is from the book "Introduction to Commutative Algebra" by Atiyah & Macdonald. I have the following two questions:

1. Can someone make me understand how the submodule $D$ is defined? since the element $x_i$ is in $M_i$ and and $u_ij (x_i)$ is in $M_j$, therefore I can't get how can we subtract these two elements.

2. How the arbitrary element of the quotient $M=C/D$ will look like?

Answer 1

1. Just before the highlighted part, $C$ is defined to be the direct sum of all $M_i$'s, and the natural inclusions $\iota_i:M_i\hookrightarrow C$ are implicitly applied here: to say it rigorously, $D$ is the submodule generated by the elements $\iota_i(x_i) - \iota_j(\mu_{ij}(x_i))$.

2. So that, in $C/D$ we will have all elements of each $M_i$ present, and (for their equivalence classes), $x_i=\mu_{ij}(x_j)$.
   An arbitrary element of $C$ is of the form $x_{i_1}+\dots +x_{i_k}$ with $k\in\Bbb N, i_j\in I$.
   Since $I$ is assumed to be directed, and based on the above equality, there's an index $j\ge i_1,\dots, i_k$ and an element $y_j\in M_j$ so that in $C/D$, $$x_{i_1}+\dots +x_{i_k}=y_j$$

3. Try to simplify it for the case when each $\mu_{ij}$ is an embedding (and, say, $I=\Bbb N$).