Module is semisimple if and only if M is the sum of simple submodules.

Question

In Lam's book, I read:

A left $R$-module $M$ is semisimple if and only if $M$ is a direct sum of simple submodules.

Questions:

(1) Is this a finite direct sum? If not, how do we define the infinite direct sum? I.e., how do we define

$$\bigoplus_{i \in I} M_i$$ where $M_i$ are submodules of $M$ and $I$ is infinite. I guess we could define it as

$$\bigoplus_{i \in I} M_i := \bigcup_{J \subseteq I, |J|< \infty}\bigoplus_{j \in J} M_j$$

Answer 1

The union you wrote naively doesn't make sense: it's not a priori clear that $\oplus_{j \in J} M_j$ is a subspace of some larger module. But it is true in any interpretation that makes sense.

The direct sum $\bigoplus_{i \in I} M_i$ of modules makes sense whether $I$ is finite or infinite: it is the unique module with maps $M_i \to \bigoplus_{i \in I} M_i$ such that maps $f_i: M_i \to N$ define a unique map $\oplus f_i: \bigoplus_{i \in I} M_i \to N$. (This is an example of a categorical colimit.) The direct sum of modules may be constructed as the submodule of $\prod_{i \in I} M_i$ of sequences with only finitely many nonzero entries. From this construction, it is clear that your union formula makes sense.