Definitions:
$1)$ Let $M_{\alpha}$, $\alpha \in I$ be a family of submodules of a module $N$ over a Ring $R$. Then we define the sum of the modules
$$\sum_{\alpha \in I}M_{\alpha}= \{x_{\alpha _{1}}+x_{\alpha _{2}}+…+x_{\alpha _{k}}:x_{\alpha _{i}}\in M_{\alpha _{i}},\enspace\alpha _{i}\in I\;\text{ for }\;1\leq i\leq k \;\text{ and }\; k\in \mathbb{N} \}$$
$2)$ Let $M_{\alpha}$, $\alpha \in I$ be a family of modules over a ring $R$. We define the product of modules as
$$\Pi_{\alpha \in I}M_{\alpha}= \{(x_{\alpha}):x_{\alpha} \in M_{\alpha}, \enspace \alpha \in I \}$$
$3)$ Let $M_{\alpha}$, $\alpha \in I$ be a family of modules over a ring $R$. The direct sum of modules is the submodule of $\Pi_{\alpha \in I}M_{\alpha}$ defined as
$$\oplus_{\alpha \in I}M_{\alpha}= \{(x_{\alpha})\in \Pi_{\alpha \in I}M_{\alpha}: x_{\alpha}=0 \;\text{for all but finitely many } \alpha \in I\}$$
Theorem:
Let $N_{i}$, $i \in I$ be a family of submodules of a module $M$ over a Ring $R$. Then the following are equivalent.
$(a)$ The map $\pi:\oplus_{i \in I}N_{i}\rightarrow \sum_{i \in I}N_{i}$ given by $\pi(\{n_{i}\}_{i\in I})=\sum_{i \in I}n_{i}$ is an isomorphism.
$(b)$ $N_{i}\cap \sum_{j \in I \smallsetminus \{i\}}N_{j}=0$ for all $i \in I$.
$(c)$ Every element $x\in \sum_{i \in I}N_{i}$ can be uniquely written as $x=x_{i_{1}}+x_{i_{2}}+…+x_{i_{k}}$ where $i_{1},i_{2},…,i_{k}$ are $k$
distinct elements of $I$ and $x_{i_{j}}\in N_{i_{j}}$, $j=1,2,…,k$.
My Question:
In the Theorem, I suspect that the statement $(a)$ does not hold true due the following reasoning:
Let $I=\mathbb{N}$
Choose two specific elements from $\oplus_{i \in I}N_{i}$ i.e. $(n_{1},n_{2},…,n_{j},0,0,0,…)$ and $(0,n_{2},n_{3},…,n_{l},0,0,0,…)$.
Now, $\pi((n_{1},n_{2},…,n_{j},0,0,0,…))= n_{2}+n_{3}+0+0+0=0+n_{2}+n_{3}+0+0=\pi((0,n_{2},n_{3},…,n_{l},0,0,0,…))$.
Can anyone take a look at it? please help me to understand the above theorem and point out the mistake in my reasoning.
Thanks.
Note: Here Modules, means left modules over a Ring $R$ with $1\neq 0$.
Best Answer
You are not using the function $π$ properly. $π((n_{1},n_{2},…,n_{j},0,0,0,…))=n_{1}+n_{2}+⋯+n_{j}$.
The function $π$ is simply saying that you sum all the nonzero components in the tuple as above. The reason why you can only have finitely many nonzero elements is that if you had an infinite number of nonzero elements the sum is not defined (you do not need to worry about convergence!). The statement $(a)$ is just saying that the external direct sum is isomorphic to the internal direct sum.