Understanding the decomposition of a module in a sum simple modules

Question

I have the following property for modules:

Given a right module $M$ and a family of simple submodules $\{N_i:i \in I\}$ such that $\sum_i N_i = M$ there exists a subset $J \subseteq I$ such that $M = \oplus_{j \in J} N_j$.

My question is double:

1. Until this, for me $\oplus_{j \in J} N_j$ means tuples $(n_j)_{j \in J}$ where almost all $n_j$ are zero, but how is this set inside $M$ (perhaps this is an isomorphism?)?
2. Can you provide me examples of this theorem in the usual abelian groups or vector spaces?

Answer 1

For your first question, this is the notion of ``internal direct sum,'' which means that $N_j\subset M$, $\sum_{j}N_j=M$, and $\left(\sum_{j\neq k}N_j\right)\cap N_k=\{0\}$ for all $k$.

An example would be for $M=\mathbb{Z}^2$ and the submodules $N_1=\mathbb{Z}e_1$, $N_2=\mathbb{Z}(e_1+e_2)$, and $N_3=\mathbb{Z}e_2$.

Clearly, $M=\sum_iN_i$, but the sum is not direct (why?). However, any choice of two of the $N_i$ is direct: $$M=N_1\oplus N_2=N_1\oplus N_3=N_2\oplus N_3.$$

Answer 2

For vector space if $V=\sum_{i\in I} V_i$, there exists $J$ such that $\oplus_{i\in J}V_i=V$. This is a classical result which says that a generating family contains a basis of the vector space.