"uniquely written" definition

Question

I'm having troubles with this definition:

My problem is with the uniquely part, for example the zero element:

$0=0+0$,

but $0=0+0+0$

or $0=0+0+0+0+0+0$.

Another example, if $m \in \sum_{i=1}^{10} G_i$ and $m=g_1+g_2$, with $g_1\in G_1$ and $g_2\in G_2$,

we have: $m=g_1+g_2$ or $m=g_1+g_2+0+0$.

It seems they can't be unique!

I really need help.

Thanks a lot.

Answer 1

Well notice what the definition says. It says that for each $m \in M$, you need to be able to write $m= \sum\limits_{\lambda \in \Lambda} g_{\lambda}$ where this sum is over all $\lambda$. So for $0$, the only possibility is a sum of $0$ $\lambda$-many times.

Answer 2

The collection of nonzero elements is unique. In the case of the element $0$ the set of nonzero elements summing to zero is the empty set.