Based upon these two older questions: Show a direct product is not finitely generated. and $R^\mathbb{N}$ is not finitely generated as an $R$-module, I would like to know the answer to the following questions.
- Given an infinite family $\{M_i\}_{i \in I}$ of non-trivial left $R$-modules (where $R$ is a ring), can $\prod M_i$ be a finitely generated $R$-module ?
- Given an infinite family $\{M_i\}_{i \in I}$ of non-trivial left $R$-modules (where $R$ is a ring), can $\bigoplus M_i$ be a finitely generated $R$-module ?
It is suggested in the second link to prove that $\prod_{i \in P_n} M_i$ ($P_n$ is a subset of $I$ of cardinality $n$) requires at least $n$ generators.
Best Answer
For direct sums, this is easy. By definition, any element of $\bigoplus M_i$ has only finitely many nonzero coordinates. So, if you have finitely many elements, there are only finitely many coordinates on which any of them are nonzero, and the same is true of the entire submodule they generate. So, since infinitely many of the $M_i$ are nontrivial, that submodule cannot be all of $\bigoplus M_i$.
Surprisingly, although direct products are "bigger" than direct sums, it actually is possible for an infinite direct product of nontrivial modules to be finitely generated. For instance, let $V$ be an infinite-dimensional vector space and let $R$ be the ring of endomorphisms of $V$. Then $V$ is a left $R$-module in the obvious way. If you pick a basis $B$ for $V$, then the infinite product $V^B$ is actually isomorphic as an $R$-module to the cyclic module $R$, by sending each map $B\to V$ to its unique extension to a linear map $V\to V$.
For another class of examples (which includes some commutative rings), take any infinite family $(R_i)$ of nonzero rings and let $R=\prod R_i$. Then each $R_i$ can be considered as an $R$-module via the projection map, and the product of these $R$-modules is the cyclic module $R$.