What does this Module notation mean?

Question

I have to prove that if $\{x_i\}_{i\in I}$ is a base of a A-module M, then: $$M=\underset{i\in I}{\oplus}Ax_i=\underset{i\in I}{\oplus}(x_i)$$

Am I right to assume that I have to prove that $\sum_{i\in I} M_i = \underset{i\in I}{\oplus}Ax_i$ and so on? What would be $Ax_i$? I just started to study Free Modules, and it seems A is the ring and $x_i$ is an element of $M$. $\oplus$ denotes the direct sum, it seems.

Answer 1

Am I right to assume that I have to prove that $\sum_{i\in I} M_i = \underset{i\in I}{\oplus}Ax_i$ and so on?

It depends... what is $M_i$ supposed to be?

What would be $Ax_i$?

It is defined to be $\{ax_i\mid a\in A\}$, which is a left ideal of $A$.

$\underset{i\in I}{\oplus}Ax_i=\underset{i\in I}{\oplus}(x_i)$

$Ax_i=(x_i)$ if $A$ is commutative, which you haven't mentioned. Either way there is no point in using the notation $(x_i)$ since it is no more useful than $Ax_i$.

If you begin knowing $Ax_i$ is a left submodule of $M$, then you just have to show that $M$ is the direct sum of the $Ax_i$ via the axioms. The problem is that we have no idea what axioms for the direct sum you are working with.

If your axioms are "$\sum Ax_i=M$ and each element is expressed uniquely" then it falls right out from the definition of what a basis is.

If your axioms are "$\sum Ax_i=M$ and $(\sum_{j\neq i} Ax_j)\cap Ax_i=\{0\}$ for all $i$" then you have slightly more work to do, but it is not far from your reach.