How can I understand this construction of a surjective $R$-module homomorphism?

Question

I have the following construction

A system of generators $(x_i)_{i\in I}$ of an $R$-module $M$ induces a surjective $R$-module homomorphism $$\bigoplus_{i\in I}R\rightarrow M,~~e_i\mapsto x_i$$

I have the following questions

1. How can we think about a direct sum of a ring $R$? So it's long time ago since I have used this notation but here I don't know how elements look like.
2. Is $\bigoplus_{i\in I}R$ a module?
3. what are the $e_i$?

Thanks for your help.

Answer 1

An example may illustrate it well.

Let's say $M$ is generated by three elements $\{x_1, x_2, x_3\}$. Then the set $I$ is $\{1, 2, 3\}$ and the direct sum $\bigoplus_{i \in I}R$ is nothing but $R^3$, namely the set of all triples of elements of $R$: $$R^3 = \{(r_1, r_2, r_3): r_1, r_2, r_3 \in R\}.$$ The elements $e_i$ refer to the canonical basis elements of $R^3$, namely \begin{eqnarray}e_1&=&(1, 0, 0)\\e_2&=&(0, 1, 0)\\e_3&=&(0, 0, 1)\end{eqnarray}as elements in $R^3$. You can verify that the homomorphism $R^3 \rightarrow M$ sending $e_i$ to $x_i$ is surjective if and only if $x_1, x_2, x_3$ generate $M$.

In the general case, you just replace $I$ with another index set (which can be infinite) and the construction is similar.