How is an infinite direct sum defined?

Question

I am reading about infinite direct sums and I just need some clarification. To say that a finite sum of say modules is direct we want to show that the intersection of all those finite modules is 0. Is the definition analogous for infinite direct sums? As in an infinite sum of modules is direct iff the infinite intersection of all the modules is 0?

Answer 1

No. There is a difference between the internal direct sum of submodules, and the (external) direct sum of modules.

The direct sum of a family of modules $(M_i)_{i\in I}$ is the submodule of $\prod_{i\in I} M_i$ made up of the families $(m_i)_{i\in I}$ such that each component $m_i\in M_i$, with finite support, i.e. the set $\{i\in I\mid m_i\ne0\}$ is finite.

The internal direct sum of a family of submodules $(M_i)_{i\in I}$ is the sum $\;\sum_{i\in I} M_i$ of these submodules which is isomorphic to tdirect sum of the $M_i$ in the previous sense.

A criterion to be an internal direct sum is that, for each $i\in I$, $$M_i\cap \sum_{\substack{j\in I\\j\ne i}} M_j=\{0\}.$$

Answer 2

In a direct sum, every element is a unique sum of finitely many elements from the summands.
As the comment says, not only the intersection of all modules have to be zero, but they must be also pairwise disjoint.

However, neither being pairwise disjoint does suffice, even for a finite ($>2$) number of modules: consider e.g. $3$ lines as submodules in ${}_{\Bbb R}\Bbb R^2$.

Let $M_i$ be submodules of $M$. The general condition for their generated submodule $\langle M_i\rangle$ to be their internal direct sum is that for each $i$ we have $$M_i\cap\langle M_j\rangle_{j\ne i}\, =\, \{0\} \,. $$

There is also the external definition as $$\bigoplus_i M_i\, :=\, \{(m_i)_i\mid m_i=0\ \text{except for finitely many}\ i \} \, \subseteq \prod_i M_i $$ You can prove that it is isomorphic to the internal direct product once all $M_i$ are embedded in a common ambient module, satisfying the above criterium.

Answer 3

An internal direct sum consists of (finite) sums of elements drawn from component submodules with the requirement that summands (elements) are uniquely determined.

Making an infinite direct sum then means having infinitely many component submodules, of which only finitely many are needed to express any one particular element of the direct sum. Effectively the expressions using additional components will consist of adding zeroes from those submodules, so the uniqueness of expressions implies a component has trivial intersection with the sum of all other components.